2 newton's law for rotational motion. Newton's second law for rotational motion. Steiner's theorem. The law of addition of moments of inertia

Physics

Law of conservation of angular momentum. Conditions for the equilibrium of bodies

Newton's law for rotational motion. Newton's second law for a particle moving under the action of a force F, can be written as:

Where p=mv is the momentum of the particle. Multiply this equation vectorially by the radius vector of the particle r. Then

(18.1)

We now introduce new quantities - angular momentum L = rp And moment of power N = rF. Then the resulting equation takes the form:

For a particle making a circular motion in a plane (x, y), the angular momentum vector is directed along the axis z(i.e. along the angular velocity vector w) and is equal in modulo

(18.3)

Let's introduce the notation: I = m r 2. Value I is called the moment of inertia of a material point about the axis passing through the origin. For a system of points rotating around an axis z with the same angular velocity, one can generalize the definition of the moment of inertia by taking the sum of the moments of inertia of all points about a common axis of rotation: I = a m i r i 2. Using the concept of an integral, one can also determine the moment of inertia of an arbitrary body about the axis of rotation. In any case, it can be written that the angular momentum vector of a system of points or a body rotating with the same angular velocity around a common axis is equal to

Then the equation of motion of a body rotating around some axis takes the form:

Here is the moment of force N- a vector directed along the axis of rotation and equal in absolute value to the product of the modulus of force and the distance along the perpendicular from the point of application of the force to the axis of rotation (shoulder of the force).

Conservation of angular momentum in the field of central forces. If the force acting on the body from another body (located at the origin) is always directed along the radius vector r connecting these bodies, then it is called the central force. In this case, the vector product r F is equal to zero (as a vector product of collinear vectors). Therefore, the moment of force is equal to zero N and the equation of rotational motion takes the form dl/dt = 0. This implies that the vector L does not depend on time. In other words, in the field of central forces, the angular momentum is conserved.

The statement proved for one particle can be extended to a closed system containing an arbitrary number of particles. Thus, in a closed system where central forces act, the total angular momentum of all particles is conserved.

So, in an arbitrary closed conservative mechanical system, in the general case, there are seven conserved quantities - energy, three components of momentum and three components of angular momentum, which have the property that for a system of particles the values of these quantities represent the sum of the values taken for individual particles. In other words, the total energy of the system is equal to the sum of the energies of individual particles, and so on.

Statics. The section of mechanics that studies the conditions for the equilibrium of extended, absolutely rigid bodies is called statics. The body is called absolutely solid if the distance between any pair of its points is constant. By definition, a body is in a state of static equilibrium if all points of the body are at rest in some inertial frame of reference.

The first equilibrium condition in ISO: the sum of all external forces applied to the body is zero.

In this case, the acceleration of the center of inertia (center of mass) of the body is zero. One can always find a frame of reference in which the center of inertia is at rest.

However, this condition does not mean that all points of the body are at rest. They can take part in rotational motion around some axis. Therefore, there is a second equilibrium condition in the ISO: the sum of the moments of all external forces about any axis is zero.

A rigid body rotating around some axes passing through the center of mass, if it is freed from external influences, maintains rotation indefinitely. (This conclusion is similar to Newton's first law for translational motion).

The occurrence of rotation of a rigid body is always caused by the action of external forces applied to individual points of the body. In this case, the appearance of deformations and the appearance of internal forces are inevitable, which in the case of a solid body ensure the practical preservation of its shape. When the action of external forces ceases, the rotation is preserved: internal forces can neither cause nor destroy the rotation of a rigid body.

The result of the action of an external force on a body with a fixed axis of rotation is an accelerated rotational motion of the body. (This conclusion is similar to Newton's second law for translational motion).

The basic law of the dynamics of rotational motion: in an inertial frame of reference, the angular acceleration acquired by a body rotating about a fixed axis is proportional to the total moment of all external forces acting on the body, and inversely proportional to the moment of inertia of the body about a given axis:

It is possible to give a simpler formulation the basic law of the dynamics of rotational motion(also called Newton's second law for rotational motion): the torque is equal to the product of the moment of inertia and the angular acceleration:

angular momentum(angular momentum, angular momentum) of a body is called the product of its moment of inertia times the angular velocity:

The angular momentum is a vector quantity. Its direction coincides with the direction of the angular velocity vector.

The change in angular momentum is defined as follows:

. (I.112)

A change in the angular momentum (with a constant moment of inertia of the body) can occur only as a result of a change in the angular velocity and is always due to the action of the moment of force.

According to the formula, as well as formulas (I.110) and (I.112), the change in angular momentum can be represented as:

. (I.113)

The product in formula (I.113) is called impulse moment of force or driving moment. It is equal to the change in angular momentum.

Formula (I.113) is valid provided that the moment of force does not change with time. If the moment of force depends on time, i.e. , then

. (I.114)

Formula (I.114) shows that: the change in angular momentum is equal to the time integral of the moment of force. In addition, if this formula is presented in the form: , then the definition will follow from it moment of force: the instantaneous moment of force is the first derivative of the moment of momentum with respect to time,

angular momentum(angular momentum, angular momentum) of a body is called the product of its moment of inertia times its angular velocity:

The angular momentum is a vector quantity. Its direction coincides with the direction of the angular velocity vector.

The change in angular momentum is defined as follows:

A change in the angular momentum (with a constant moment of inertia of the body) can occur only as a result of a change in the angular velocity and is always due to the action of the moment of force.

According to the formula, as well as formulas (I.110) and (I.112), the change in angular momentum can be represented as:

The product in formula (I.113) is called impulse moment of force or driving moment. It is equal to the change in angular momentum.

Formula (I.113) is valid provided that the moment of force does not change with time. If the moment of force depends on time, i.e. , then

Expression (I.115) is another form master equation (law ) dynamics of rotational motion of a rigid body relative to the fixed axis: the derivative of the angular momentum of a rigid body with respect to an axis is equal to the moment of forces with respect to the same axis.

In a closed system, the moment of external forces and, therefore:

Formula (I.116) is law of conservation of angular momentum: the vector sum of all angular momenta about any axis for a closed system remains constant in the case of equilibrium of the system. In accordance with this, the angular momentum of a closed system with respect to any fixed point does not change with time Law of conservation of angular momentum - fundamental law of nature.

Please note: the total angular momentum of the system is equal to the vector sum of the angular momentum of the individual parts of the system.

Application of Newton's second law for rotational motion

According to Newton's second law, the acceleration of a body under the action of a force is proportional to the magnitude of the force and inversely proportional to the mass of the object:

Let's ask ourselves, does Newton's second law apply to rotational motion?

Using analogues of the characteristics of translational and rotational motions, Newton's second law for rotational motion will have the form:

the role of acceleration a is performed by the angular acceleration α;
the role of force F is the moment of force M;
mass m - replaces the moment of inertia I.

Let us assume that the body moves in a circle under the action of a tangential force applied tangentially to the circle, which leads to an increase in the tangential velocity of the ball, not to be confused with a normal force directed along the radius of the circle of rotation (tangential and normal speeds are discussed in detail on the page "Parameters of rotational motion" ).

Let's multiply both sides of the equation describing Newton's second law by the radius of the circle r:

Thus, we have made the transition from Newton's second law for translational motion to its counterpart for rotational motion. It should be noted that this formula is valid only for a material point; for an extended object, it is necessary to use other formulas that will be discussed later.

To complete the transition from the description of translational to rotational motion, we use the relationship between angular acceleration α and tangential acceleration a:

Substituting one formula into another, we get:

The resulting formula connects the moment of force acting on a material point and its angular acceleration. Communication is carried out through the coefficient of proportionality m r 2, which is called moment of inertia material point and denote I (measured in kg m 2).

As a result, we got the equivalent of Newton's second law for rotational motion:

In the event that several forces act simultaneously on the body, Newton's second law takes the following form:

ΣF is the vector sum of all forces that act on the object.

If several moments of forces act simultaneously on an object, Newton's second law will take the form:

ΣM is the vector sum of all moments of forces that act on the object.

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1. Write the basic equation for the dynamics of rotational motion (Newton's 2nd law for rotational motion).

This expression is called the basic equation of the dynamics of rotational motion and is formulated as follows: the change in the angular momentum of a rigid body is equal to the momentum momentum of all external forces acting on this body.

2. What is the moment of force? (formula in vector and scalar form, figures).

Moment strength (synonyms: torque; rotational moment; torque) is a physical quantity that characterizes the rotational action of a force on a rigid body.

Moment of force - vector quantity (M?)

(vector view) М?= |r?*F?|,r– distance from the axis of rotation, to the point of force application.

(sort of like a scalar view) |M|=|F|*d

The vector of the moment of force - coincides with the axis O 1 O 2, its direction is determined by the rule of the right screw. The moment of force is measured in newton meters. 1 N m - the moment of force that produces a force of 1 N on a lever 1 m long.

3. What is called a vector: rotation, angular velocity, angular acceleration. Where are they directed, how to determine this direction in practice?

Vectors are pseudovectors or axial vectors that do not have a specific application point: they are plotted on the rotation axis from any of its points.

Angular movement is a pseudovector, the module of which is equal to the angle of rotation, and the direction coincides with the axis around which the body rotates, and is determined by the rule of the right screw: the vector is directed in the direction from which the rotation of the body is visible counterclockwise (measured in radians)

Angular velocity is a value that characterizes the speed of rotation of a rigid body, equal to the ratio of the elementary angle of rotation and the elapsed time dt, during which this rotation took place.

Angular velocity vector is directed along the axis of rotation according to the rule of the right screw, just like the vector.

Angular acceleration is a value that characterizes the speed of movement of the angular velocity.

The vector is directed along the axis of rotation towards the vector during accelerated rotation and opposite to the vector during slow rotation.

4. How does a polar vector differ from an axial one?

Polar vector has a pole and axial- No.

5. What is called the moment of inertia of a material point, a rigid body?

Moment inertia- the value characterizing the measure of inertia material points as it rotates around an axis. Numerically, it is equal to the product of the mass and the square of the radius (distance to the axis of rotation). For solid body moment of inertia is equal to the sum of the moments of inertia of its parts, and therefore can be expressed in integral form:

6. What parameters does the moment of inertia of a rigid body depend on?

From geometric dimensions

From the choice of the axis of rotation

7. Steiner's theorem (explanatory figure).

Theorem: the moment of inertia of a body about an arbitrary axis is equal to the sum of the moment of inertia of this body about an axis parallel to it, passing through the center of mass of the body, and the product of the body's mass by the square of the distance between the axes:

- the desired moment of inertia about the parallel axis

is the known moment of inertia about the axis passing through the center of mass of the body

- distance between the indicated axes

8. Moment of inertia of a ball, cylinder, rod, disk.

Moment of inertia m.t. relative to the pole is called a scalar quantity equal to the product of this mass. points per square of the distance to the pole..

Moment of inertia m.t. can be found using the formula

where m is the mass of the b.w., R is the distance to the pole 0.

The unit of moment of inertia in SI is the kilogram multiplied by the meter squared (kg m 2).

1.Straight thin rod length l and the masses m

1) The axis is perpendicular to the rod and passes through its center of mass

2) The axis is perpendicular to the rod and passes through its end

2.Ball radius r and the masses m

The axis passes through the center of the ball

3.Hollow thin-walled cylinder or radius ring r and the masses m

4.Solid cylinder or radius disc r and the masses m

5.Solid length cylinder l, radius r and the masses m

The axis is perpendicular to the cylinder and passes through its center of mass

9.How to determine the direction of the moment of force?

The moment of force about some point is the cross product strength on the shortest distance from this point to the line of action of the force.

M- moment of force (Newton meter), F- Applied force (Newton), r- distance from the center of rotation to the place of application of force (meter), l- the length of the perpendicular dropped from the center of rotation to the line of action of the force (meter), ? is the angle between the force vector F and position vector r

Moment of power - axial vector. It is directed along the axis of rotation. The direction of the vector of the moment of force is determined by the gimlet rule, and its magnitude is equal to M.

10. How are the moment of forces, angular velocities, moments of impulse added up?

If several forces act simultaneously on a body that can rotate around a point, then to add the moments of these forces, the rule of adding the moments of forces should be used.

The rule for adding the moments of forces reads - The resulting vector of the moment of force is equal to the geometric sum of the component vectors of the moments with

For the rule of addition of moments of forces, two cases are distinguished

1. The moments of forces lie in the same plane, the axes of rotation are parallel. Their sum is determined by algebraic addition. Right-handed moments are included in the sum with a sign minus. Left hand screw - with sign plus

2. The moments of forces lie in different planes, the axes of rotation are not parallel. The sum of moments is determined by geometric addition of vectors.

Angular velocity (rad / s) - a physical quantity, which is an axial vector and characterizes the speed of rotation of a material point around the center of rotation. The angular velocity vector is equal in magnitude to the angle of rotation of the point around the center of rotation per unit time

is directed along the axis of rotation according to the rule of the gimlet, that is, in the direction in which the gimlet with a right-hand thread would be screwed if it rotated in the same direction.

Angular velocities are plotted on the axis of rotation and can be added if they are directed in one direction, in the opposite direction they are subtracted

In the International System of Units (SI), momentum is measured in kilogram meters per second (kg m/s).

Moment? nt and? pulse characterizes the amount of rotational motion. A quantity that depends on how much mass is rotating, how it is distributed about the axis of rotation, and how fast the rotation occurs.

If there is a material point with a mass moving at a speed and located at a point described by the radius vector, then the angular momentum is calculated by the formula:

where is the sign of the vector product

11. Formulate the law of conservation of total mechanical energy in relation to a body rotating around a fixed axis.

the potential energy is maximum at the initial point of the pendulum's motion. The potential energy of MgH turns into kinetic energy, which is maximum at the moment the pendulum lands on the ground.

Io-moment of inertia about the axis for one weight (we have 4 of them)

I= 4Io=4ml^2 (Io=ml^2)

12. Formulate the law of conservation of total mechanical energy in relation to a body rotating around a fixed axis.

The angular momentum of a rotating body is directly proportional to the speed of rotation of the body, its mass and linear extent. The higher any of these values, the higher the angular momentum.

In mathematical representation, the angular momentum L a body rotating at an angular velocity ? , is equal to L=I?, where the value I called moment of inertia

the speed of rotation of the pendulum increases many times due to a decrease in the moment of inertia while maintaining the moment of rotation. Here we see clearly that the smaller the moment of inertia I, the higher the angular velocity ? and, as a consequence, a shorter period of rotation, inversely proportional to it.

Angular moment of a rotating body

where is the body weight; - speed; is the radius of the orbit along which the body moves; - moment of inertia; is the angular velocity of the rotating body.

Law of conservation of angular momentum:

– for rotational movement

13. What expression determines the work of the moment of forces

In the SI system, work is measured in Joules, moment of force in Newton * meter, and ANGLE in radians

Usually known is the angular velocity in radians per second and the duration of the TORQUE.

Then the WORK done by the TORQUE of force is calculated as:

14. Get a formula that determines the power developed by the moment of forces.

If a force performs an action at any distance, then it performs mechanical work. Also, if a moment of force performs an action through an angular distance, it does work.

In the SI system, power is measured in watts, torque in Newton meters, and ANGULAR VELOCITY in radians per second.

Of course, the position of one, even "special", point does not completely describe the motion of the entire system of bodies under consideration, but still it is better to know the position of at least one point than not to know anything. Nevertheless, let us consider the application of Newton's laws to the description of the rotation of a rigid body about a fixed axis 1 .

Let's start with the simplest case: let the material point of the mass m attached with a weightless rigid rod of length r to the fixed axis OO /(Fig. 106).

A material point can move around the axis, remaining at a constant distance from it, therefore, its trajectory will be a circle centered on the axis of rotation.

Of course, the motion of a point obeys the equation of Newton's second law

However, the direct application of this equation is not justified: firstly, the point has one degree of freedom, so it is convenient to use the rotation angle as the only coordinate, and not two Cartesian coordinates; secondly, the reaction forces in the axis of rotation act on the system under consideration, and directly on the material point - the tension force of the rod. Finding these forces is a separate problem, the solution of which is redundant for describing rotation. Therefore, it makes sense to obtain, on the basis of Newton's laws, a special equation that directly describes the rotational motion.

Let at some point in time a certain force acts on a material point F, lying in a plane perpendicular to the axis of rotation (Fig. 107).

In the kinematic description of curvilinear motion, the total acceleration vector a is conveniently decomposed into two components, the normal a n, directed to the axis of rotation, and tangential and τ directed parallel to the velocity vector. We do not need the value of normal acceleration to determine the law of motion. Of course, this acceleration is also due to acting forces, one of which is the unknown tensile force on the rod.

Let us write the equation of the second law in the projection onto the tangential direction:

Note that the reaction force of the rod is not included in this equation, since it is directed along the rod and perpendicular to the selected projection. Changing the angle of rotation φ directly determined by the angular velocity

the change of which, in turn, is described by the angular acceleration

Angular acceleration is related to the tangential acceleration component by the relation

If we substitute this expression into equation (1), we obtain an equation suitable for determining the angular acceleration. It is convenient to introduce a new physical quantity that determines the interaction of bodies during their rotation. To do this, we multiply both sides of equation (1) by r:

Consider the expression on its right side F r that makes sense

the product of the tangential component of the force and the distance from the axis of rotation to the point of application of the force. The same work can be presented in a slightly different form (Fig. 108):

here d is the distance from the axis of rotation to the line of action of the force, which is also called the shoulder of the force.

This physical quantity is the product of the modulus of force and the distance from the line of action of the force to the axis of rotation (arm of force) M = Fd− is called the moment of force. The action of a force can result in both clockwise and counterclockwise rotation. In accordance with the chosen positive direction of rotation, the sign of the moment of force should also be determined. Note that the moment of force is determined by the component of the force that is perpendicular to the radius vector of the point of application. The component of the force vector directed along the segment connecting the point of application and the axis of rotation does not lead to untwisting of the body. This component, when the axis is fixed, is compensated by the reaction force in the axis, therefore it does not affect the rotation of the body.

Let's write down one more useful expression for the moment of force. Let the power F attached to a point BUT, whose Cartesian coordinates are X, at(Fig. 109).

Let's decompose the force F into two components F x, F, parallel to the corresponding coordinate axes. The moment of force F about the axis passing through the origin is obviously equal to the sum of the moments of the components F x, F, i.e

Similarly, the way we introduced the concept of the vector of angular velocity, we can also define the concept of the vector of the moment of force. The module of this vector corresponds to the definition given above, but it is directed perpendicular to the plane containing the force vector and the segment connecting the point of application of the force with the axis of rotation (Fig. 110).

The vector of the moment of force can also be defined as the vector product of the radius vector of the point of application of the force and the force vector

Note that when the point of application of force is displaced along the line of its action, the moment of force does not change.

Let us denote the product of the mass of a material point by the square of the distance to the axis of rotation

(this value is called the moment of inertia of a material point about the axis). Using these notations, equation (2) takes on a form that formally coincides with the equation of Newton's second law for translational motion:

This equation is called the basic equation of rotational motion dynamics. So, the moment of force in rotational motion plays the same role as the force in translational motion - it is he who determines the change in angular velocity. It turns out (and this is confirmed by our everyday experience) that the influence of force on the speed of rotation is determined not only by the magnitude of the force, but also by the point of its application. The moment of inertia determines the inertial properties of the body with respect to rotation (in simple terms, it shows whether it is easy to spin the body): the farther from the axis of rotation a material point is, the more difficult it is to bring it into rotation.

Equation (3) can be generalized to the case of rotation of an arbitrary body. When a body rotates around a fixed axis, the angular accelerations of all points of the body are the same. Therefore, just as we did when deriving Newton's equation for the translational motion of a body, we can write equations (3) for all points of a rotating body and then sum them up. As a result, we obtain an equation that outwardly coincides with (3), in which I- the moment of inertia of the whole body, equal to the sum of the moments of its constituent material points, M is the sum of moments of external forces acting on the body.

Let us show how the moment of inertia of a body is calculated. It is important to emphasize that the moment of inertia of a body depends not only on the mass, shape and dimensions of the body, but also on the position and orientation of the axis of rotation. Formally, the calculation procedure is reduced to dividing the body into small parts, which can be considered material points (Fig. 111),

and the summation of the moments of inertia of these material points, which are equal to the product of the mass by the square of the distance to the axis of rotation:

For bodies of a simple shape, such sums have long been calculated, so it is often enough to remember (or find in a reference book) the appropriate formula for the desired moment of inertia. As an example: the moment of inertia of a circular homogeneous cylinder, masses m and radius R, for the axis of rotation coinciding with the axis of the cylinder is equal to:

1 In this case, we restrict ourselves to considering rotation around a fixed axis, because the description of an arbitrary rotational motion of a body is a complex mathematical problem that goes far beyond the scope of a high school mathematics course. Knowledge of other physical laws, except for those considered by us, this description does not require.

Newton's law for rotational motion

The systematization of physical quantities leads to the fact that Newton's second law should not be limited to the rectilinear form of motion, but should be extended to all mechanical forms of motion, and the terminology regarding the quantities describing this law in a generalized form should be clarified.

1. Newton's second law for rectilinear motion.

Newton's second law is declared as an equation of dynamics for uneven motion in a mechanical rectilinear form of motion and is given in physics textbooks, usually in two forms:

2. Newton's second law for a rotational form of motion.

With non-uniform rotation of the body, the record of Newton's second law, similar to equation (3), should look like this:

3. Newton's second law for the orbital form of motion.

The orbital form of motion, as shown in the article on forms of motion, generally consists of 4 simple forms of motion (two rectilinear and two rotational). In the article devoted to accelerations in the orbital form of motion, equations are derived to determine the accelerations in each of these 4 forms of motion. Therefore, Newton's second law can be written for each of them in the form of equations (3) or (4).

FIt is the tangential force of inertia opposing the change in tangential velocity; m is the mass of a body moving in a circular orbit.

4. Generalized Newton's second law.

All three equations (3, 4, 5) have, as expected, the same structure, which takes into account only one generalized resistance to inertia UI, described on the page dedicated to the generalized parameters of motion forms. On this basis, it is possible to derive a generalized record of Newton's second equation in the form:

5. Dimensions and units of linear and rotational inertia.

The SI uses the unit kilogram for inertial mass, because the SI adheres to the irrelevant principle of mass equivalence. In the system of ESWL values, the linear inertia I has the dimension EL -2 T 2 and the unit J m -2 s 2 . The article devoted to the principle of mass equivalence shows that the mass in Newton's second law and the mass in the law of universal gravitation must have different dimensions and units.

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Newton's second law for rotational motion

By differentiating the angular momentum with respect to time, we obtain the basic equation for the dynamics of rotational motion, known as Newton's second law for rotational motion, formulated as follows: the rate of change of the angular momentum L a body rotating around a fixed point is equal to the resultant moment of all external forces M applied to the body, relative to this point:

Since the angular momentum of a rotating body is directly proportional to the angular velocity ? rotation, and the derivative d?/dt is the angular acceleration ? , then this equation can be represented as

where J is the moment of inertia of the body.

Equations (14) and (15), which describe the rotational motion of a body, are similar in content to Newton's second law for the translational motion of bodies ( ma = F ). As can be seen, during rotational motion as a force F moment of force is used M , as an acceleration a - angular acceleration ? , and the role of the mass m characterizing the inertial properties of the body, plays the moment of inertia J.

The moment of inertia of a rigid body determines the spatial distribution of the body's mass and is a measure of the body's inertia during rotational motion. For a material point, or an elementary mass? m i, rotating around an axis, the concept of the moment of inertia is introduced, which is a scalar quantity numerically equal to the product of the mass by the square of the distance r i to axis:

The moment of inertia of a volumetric solid is the sum of the moments of inertia of its constituent elementary masses:

For a homogeneous body with a uniformly distributed density? = ? m i /?V i (?V i– elementary volume) can be written:

or, in integral form (the integral is taken over the entire volume):

The use of equation (19) makes it possible to calculate the moments of inertia of homogeneous bodies of various shapes with respect to any axes. The simplest result, however, is obtained by calculating the moments of inertia of homogeneous symmetrical bodies about their geometric center, which in this case is the center of mass. The moments of inertia of some bodies of regular geometric shape calculated in this way relative to the axes passing through the centers of mass are shown in Table 1.

The moment of inertia of a body about any axis can be found by knowing the body's own moment of inertia, i.e. moment of inertia about an axis through its center of mass, using Steiner's theorem. According to her moment of inertia J relative to an arbitrary axis is equal to the sum of the moment of inertia J 0 about the axis passing through the center of mass of the body parallel to the considered axis, and the product of the body mass m per square distance r between axles:

The axis, during the rotation of the body around which, no moment of force arises, tending to change the position of the axis in space, is called the free axis of the given body. A body of any shape has three mutually perpendicular free axes passing through its center of mass, which are called the main axes of inertia of the body. Own moments of inertia of the body about the main axes of inertia are called the main moments of inertia.

Moments of inertia of some homogeneous bodies (with mass m) of regular geometric shape with respect to the axes passing through the centers of mass

Axis location(indicated by arrow)

hoop radius r

Disk Radius r at a thickness negligible compared to the radius

Solid cylinder radius r with height l

Hollow cylinder with inner radius r and wall thickness d

Thin rod length l

Rectangular parallelepiped with sides a, b And c

Cube with edge length a

Description of the installation and measurement principle:

The setup used in this work to study the basic regularities of the dynamics of the rotational motion of a rigid body around a fixed axis is called the Oberbeck pendulum. The general view of the installation is shown in Figure 4.

The main element of the installation, which performs rotational movement around an axis perpendicular to the plane of the figure, is a cross 1 , consisting of four screwed into the pulley 2 rods (spokes) at right angles to each other, each of which is fitted with a cylindrical load freely moving along the rod 3 mass, fixed in the desired position with a screw 4 . Along the entire length of the spokes, transverse cuts are applied at centimeter intervals, with which you can easily count the distances from the center of the location of the goods to the axis of rotation. By moving loads, a change in the moment of inertia is achieved J the entire cross.

The rotation of the crosspiece occurs under the action of the tension force (elastic force) of the thread 5 , fixed at one end in any one of the two pulleys ( 6 , or 7 ), on which, when the cross is rotated, it is wound. The other end of the string with a weight attached to it P 0 8 variable mass m 0 is thrown over a fixed block 9 , which changes the direction of the rotating tension force, coinciding with the tangent to the corresponding pulley. The use of one of two pulleys with different radii allows you to change the shoulder of the rotating force, and, consequently, its moment. M.

Verification of various patterns of rotational motion in this work is reduced to measuring the time t lowering a load from a height h.

To determine the height of the lowering of the load on the Oberbeck pendulum, a millimeter scale is used. 10 attached to a vertical post 11 . Value h corresponds to the distance between the risks, one of which is marked on the upper movable bracket 12 , and the other on the bottom bracket 13 , fixed in a rack 11 . The movable bracket can be moved along the rack and fixed in any desired position by setting the height of the load.

Automatic measurement of the time of lowering the load is carried out using an electronic millisecond watch, the digital scale of which 14 located on the front panel, and two photoelectric sensors, one of which 15 fixed on the top bracket, and the other 16 - on the lower fixed bracket. Sensor 15 gives a signal to start an electronic stopwatch at the beginning of the movement of the load from its upper position, and the sensor 16 when the load reaches the lower position, it gives a signal that stops the stopwatch, fixing the time t distance traveled by the load h, and at the same time includes located behind the pulleys 6 And 7 brake electromagnet that stops the rotation of the cross.

A simplified diagram of the pendulum is shown in Figure 5.

Per cargo P 0 constant forces act: gravity mg and the thread tension T, under the influence of which the load moves down uniformly with acceleration a. Pulley Radius r 0 under the action of the thread tension T rotates with angular acceleration?, while tangential acceleration a t extreme points of the pulley will be equal to the acceleration a descending load. Accelerations a And? related by the ratio:

If the time of lowering the load P 0 denoted by t, and the path they have traveled through h, then according to the law of uniformly accelerated motion at an initial speed equal to 0, the acceleration a can be found from the relation:

Measuring the diameter with a caliper d 0 of the corresponding pulley on which the thread is wound, and calculating its radius r o , from (21) and (22) it is possible to calculate the angular acceleration of the rotation of the cross:

When the load tied to the thread is lowered, moving with uniform acceleration, the thread unwinds and sets the flywheel in uniformly accelerated rotational motion. The force that causes the body to rotate is the tension in the thread. It can be determined from the following considerations. Since, according to Newton's second law, the product of the mass of a moving body and its acceleration is equal to the sum of the forces acting on the body, in this case, suspended on a thread and descending with uniform acceleration a body mass m 0 there are two forces: body weight m 0 g, directed downward, and the force of the thread tension T pointing up. Therefore, the following relation holds:

Therefore, the torque will be equal to:

If we neglect the force of friction of the disk on the axis of the cross, then we can assume that only the moment acts on the cross. M thread tension force T. Therefore, using Newton's second law for rotational motion (13), we can calculate the moment of inertia J crosses with loads rotating on it, taking into account (16) and (19) according to the formula:

or, substituting the expression for a (15):

The resulting equation (28) is exact. At the same time, having done experiments to determine the acceleration of the movement of the load P 0 , one can verify that a << g, and therefore in (27) the value ( g – a), neglecting the value a, can be taken equal to g. Then expression (27) will take the form:

If the quantities m 0 , r 0 and h do not change during the experiments, then there is a simple quadratic relationship between the moment of inertia of the cross and the time of lowering the load:

where K = m 0 r 0 2 g/2h. Thus, by measuring the time t weight lowering m 0 , and knowing the height of its lowering h, you can calculate the moment of inertia of the cross, consisting of the spokes, the pulley in which they are fixed, and the weights located on the cross. Formula (30) makes it possible to check the main regularities of the rotational motion dynamics.

If the moment of inertia of the body is constant, then different torques M 1 and M 2 will tell the body different angular accelerations? 1 and? 2 , i.e. will have:

Comparing these expressions, we get:

On the other hand, the same torque will give bodies with different moments of inertia different angular accelerations. Really,

Work order:

Exercise 1 . Determining the moment of inertia of the cross and checking the dependence of the angular acceleration on the moment of the rotating force.

The task is carried out with a crosspiece without weights put on it.

Select and set height h lowering the load m 0 by moving the upper movable bracket 12 (height h may be assigned by the teacher). Meaning h enter in table 2.

Measure the diameter of the selected pulley with a caliper and find its radius r 0 . Meaning r 0 enter in table 2.

By choosing the smallest value of the mass m 0 , equal to the mass of the stand on which additional weights are put on, wind the thread around the selected pulley so that the load m 0 was elevated h. Measure three times the time t 0 lowering this load. Record the data in table 2.

Repeat the previous experiment, for different (from three to five) masses m 0 of the descending load, taking into account the mass of the stand on which the loads are put on. The masses of the stand and weights are indicated on them.

After each experiment, carry out the following calculations (entering their results in table 2):

calculate the average time of lowering the load t 0 Wed and, using it, by formula (22) determine the linear acceleration of the loads a. Points on the surface of the pulley move with the same acceleration;

knowing the radius of the pulley r 0 , using formula (23) find its angular acceleration?;

using the obtained value of linear acceleration a using formula (26) find the torque M;

based on the received values? And M calculate by formula (29) the moment of inertia of the flywheel J 0 without weights on the rods.

Based on the results of all experiments, calculate and enter in table 2 the average value of the moment of inertia J 0, avg. .

For the second and subsequent experiments, calculate, entering the calculation results in Table 2, the relationship? i /? 1 and M i / M 1 (i is the number of experience). Check if the ratio is correct M i / M 1 = ? 1 /? 2 .

According to Table 2, for any one line, calculate the measurement errors of the moment of inertia using the formula:

Values of absolute errors? r, ?t, ?h consider equal to instrumental errors; ? m 0 = 0.5 g

Installation parameters constant in this task, used in calculations:

Rotational movement of the body. Law of Rotational Motion

This article describes an important section of physics - "Kinematics and dynamics of rotational motion."

Basic concepts of kinematics of rotational motion

The rotational movement of a material point around a fixed axis is such a movement, the trajectory of which is a circle located in a plane perpendicular to the axis, and its center lies on the axis of rotation.

The rotational motion of a rigid body is a motion in which all points of the body move along concentric (the centers of which lie on the same axis) circles in accordance with the rule for the rotational motion of a material point.

Let an arbitrary rigid body T perform rotations around the axis O, which is perpendicular to the plane of the figure. Let us choose a point M on the given body. During rotation, this point will describe a circle around the O axis with a radius r.

After some time, the radius will rotate relative to its original position by an angle Δφ.

The direction of the right screw (clockwise) is taken as the positive direction of rotation. The change in the angle of rotation with time is called the equation of rotational motion of a rigid body:

If φ is measured in radians (1 rad is the angle corresponding to an arc with a length equal to its radius), then the length of the circular arc ΔS, which the material point M will pass in time Δt, is equal to:

The main elements of the kinematics of uniform rotational motion

A measure of the movement of a material point in a short period of time dt serves as an elementary rotation vector dφ.

The angular velocity of a material point or body is a physical quantity, which is determined by the ratio of the elementary rotation vector to the duration of this rotation. The direction of the vector can be determined by the rule of the right screw along the O axis. In scalar form:

If ω = dφ/dt = const, then such a motion is called uniform rotational motion. With it, the angular velocity is determined by the formula

According to the preliminary formula, the dimension of the angular velocity

The uniform rotational motion of a body can be described by a period of rotation. The rotation period T is a physical quantity that determines the time during which the body around the axis of rotation performs one complete revolution ([T] = 1 s). If in the formula for the angular velocity we take t = T, φ = 2 π (full one revolution of radius r), then

Therefore, the rotation period is defined as follows:

The number of revolutions that a body makes per unit time is called the rotation frequency ν, which is equal to:

Frequency units: [ν] \u003d 1 / c \u003d 1 c -1 \u003d 1 Hz.

Comparing the formulas for the angular velocity and rotation frequency, we obtain an expression relating these quantities:

The main elements of the kinematics of non-uniform rotational motion

The uneven rotational motion of a rigid body or a material point around a fixed axis characterizes its angular velocity, which changes with time.

Vector ε characterizing the rate of change of the angular velocity is called the angular acceleration vector:

If the body rotates, accelerating, that is dω/dt > 0, the vector has a direction along the axis in the same direction as ω.

If the rotational movement is slowed down - dω/dt< 0 , then the vectors ε and ω are oppositely directed.

Comment. When an uneven rotational motion occurs, the vector ω can change not only in magnitude, but also in direction (when the rotation axis is rotated).

Relationship between quantities characterizing translational and rotational motion

It is known that the length of the arc with the angle of rotation of the radius and its value is related by the relation

Then the linear velocity of a material point performing a rotational motion

The normal acceleration of a material point that performs rotational translational motion is defined as follows:

So, in scalar form

Tangential accelerated material point that performs rotational motion

Angular moment of a material point

The vector product of the radius-vector of the trajectory of a material point with mass m i and its momentum is called the angular momentum of this point about the axis of rotation. The direction of the vector can be determined using the right screw rule.

Angular moment of a material point ( L i) is directed perpendicular to the plane drawn through r i and υ i , and forms with them the right triple of vectors (that is, when moving from the end of the vector r i to υ i the right screw will show the direction of the vector L i).

In scalar form

Considering that when moving in a circle, the radius vector and the linear velocity vector for the i-th material point are mutually perpendicular,

So the angular momentum of a material point for rotational motion will take the form

Moment of force acting on the i-th material point

The vector product of the radius-vector, which is drawn to the point of application of the force, and this force is called the moment of force acting on the i-th material point relative to the axis of rotation.

Value l i , equal to the length of the perpendicular dropped from the point of rotation to the direction of the force, is called the arm of the force F i.

Rotational dynamics

The equation for the dynamics of rotational motion is written as follows:

The formulation of the law is as follows: the rate of change of the angular momentum of a body that rotates around a fixed axis is equal to the resulting moment about this axis of all external forces applied to the body.

Moment of momentum and moment of inertia

It is known that for the i-th material point the angular momentum in scalar form is given by the formula

If instead of the linear velocity we substitute its expression in terms of the angular one:

then the expression for the angular momentum will take the form

Value I i = m i r i 2 is called the moment of inertia about the axis of the i-th material point of an absolutely rigid body passing through its center of mass. Then we write the angular momentum of the material point:

We write the angular momentum of an absolutely rigid body as the sum of the angular momentum of the material points that make up this body:

Moment of force and moment of inertia

The law of rotation says:

It is known that the angular momentum of a body can be represented in terms of the moment of inertia:

Considering that the angular acceleration is determined by the expression

we obtain the formula for the moment of force, represented through the moment of inertia:

Comment. The moment of force is considered positive if the angular acceleration by which it is caused is greater than zero, and vice versa.

Steiner's theorem. The law of addition of moments of inertia

If the body's axis of rotation does not pass through its center of mass, then its moment of inertia can be found relative to this axis using Steiner's theorem:

where I 0 is the initial moment of inertia of the body; m- body mass; a- the distance between the axles.

If the system that rotates around the fixed axis consists of n bodies, then the total moment of inertia of this type of system will be equal to the sum of the moments of its components (the law of addition of moments of inertia).

1. Write the basic equation for the dynamics of rotational motion (Newton's 2nd law for rotational motion).

This expression is called the basic equation of the dynamics of rotational motion and is formulated as follows: change in the angular momentum of a rigid body, equal to momentum momentum all external forces acting on the body.

2. What is the moment of force? (formula in vector and scalar form, figures).

Momentstrength(synonyms:torque; rotational moment; torque ) - physical quantitycharacterizing the rotational action forces on a rigid body.

Moment of force - vector quantity (М̅)

(vector view) М̅= |r̅*F̅|, r – distance from the axis of rotation, to the point of force application.

(sort of like a scalar view) |M|=|F|*d

The vector of the moment of force - coincides with the axis O 1 O 2, its direction is determined by the rule of the right screw.
The moment of force is measured in newton meters. 1 N m - moment of force , which produces a force of 1 N on a lever 1 m long.

3. What is called a vector: rotation, angular velocity, angular acceleration. Where are they directed, how to determine this direction in practice?

Vectors are pseudovectors or axial vectors that do not have a specific application point: they are plotted on the rotation axis from any of its points.

Angular movement- this is a pseudovector, the module of which is equal to the angle of rotation, and the direction coincides with the axis around which the body rotates, and is determined by the rule of the right screw: the vector is directed in the direction from which the rotation of the body is visible counterclockwise (measured in radians)
Angular velocity- a value that characterizes the speed of rotation of a rigid body, equal to the ratio of the elementary angle of rotation and the elapsed time dt, during which this rotation took place.

Angular velocity vector is directed along the axis of rotation according to the rule of the right screw, just like the vector.

Angular acceleration- a value that characterizes the speed of movement of the angular velocity.

The vector is directed along the axis of rotation towards the vector during accelerated rotation and opposite to the vector during slow rotation.

4. How is a polar vector different from an axial one?

Polarvectorhas a pole andaxial- No.

5. What is the moment of inertia of a material point, a rigid body?

Momentinertia- the value characterizing the measure inertia material points as it rotates around an axis. Numerically, it is equal to the product of the mass and the square of the radius (distance to the axis of rotation).For solid body moment of inertia is equal to the sum of the moments of inertia its parts, and therefore can be expressed in integral form:

I =∫ r 2 dү.

6. On what parameters does the moment of inertia of a rigid body depend?

From body weight
From geometric dimensions
From the choice of the axis of rotation

7. Steiner's theorem (explanatory figure).

Known moment of inertia about an axis passing through the body's center of mass

Body mass

- distance between the indicated axes

8. Moment of inertia of a ball, cylinder, rod, disk.

Moment of inertia m.t. relative to the pole is called a scalar quantity equal to the product of this mass. points per square of the distance to the pole..

Moment of inertia m.t. can be found using the formula

The axis passes through the center of the ball

Cylinder axis

The axis is perpendicular to the cylinder and passes through its center of mass
9.How to determine the direction of the moment of force?

The moment of force about some point is the cross product strength on the shortest distance from this point to the line of action of the force.

[M] = newton meter

M- moment of force (Newton meter), F- Applied force (Newton), r- distance from the center of rotation to the place of application of force (meter), l- the length of the perpendicular dropped from the center of rotation to the line of action of the force (meter), α is the angle between the force vector F and position vector r

M = Fl = F r sin(α )

(m,F,r-vector quantities)

Moment of power - axial vector. It is directed along the axis of rotation. The direction of the vector of the moment of force is determined by the gimlet rule, and its magnitude is equal to M.
10. How are the moment of forces, angular velocities, moments of impulse added up?

Moment of forces

If several forces act simultaneously on a body that can rotate around a point, then to add the moments of these forces, the rule of adding the moments of forces should be used.

The rule for adding the moments of forces reads - The resulting vector of the moment of force is equal to the geometric sum of the component vectors of the moments with

For the rule of addition of moments of forces, two cases are distinguished

2. The moments of forces lie in different planes, the axes of rotation are not parallel. The sum of moments is determined by geometric addition of vectors.

Angular velocities

Angular velocity (rad / s) - a physical quantity that is an axial vector and characterizes the speed of rotation of a material point around the center of rotation. The angular velocity vector is equal in magnitude to the angle of rotation of the point around the center of rotation per unit time

Angular velocities are plotted on the axis of rotation and can be added if they are directed in one direction, in the opposite direction they are subtracted

angular momentum

In the International System of Units (SI), momentum is measured in kilogram meters per second (kg m/s).

The angular momentum characterizes the amount of rotational motion. A quantity that depends on how much mass is rotating, how it is distributed about the axis of rotation, and how fast the rotation occurs.

If there is a material point with a mass moving at a speed and located at a point described by the radius vector, then the angular momentum is calculated by the formula:
where is the sign of the vector product

To calculate the angular momentum of a body, it must be broken into infinitesimal pieces and vector sum their moments as moments of momentum of material points, that is, take the integral:
11. Formulate the law of conservation of total mechanical energy in relation to a body rotating around a fixed axis.
MgH=(IoW^2)/2

the potential energy is maximum at the initial point of the pendulum's motion. The potential energy of MgH turns into kinetic energy, which is maximum at the moment the pendulum lands on the ground.
Io-moment of inertia about the axis for one weight (we have 4 of them)

I= 4Io=4ml^2 (Io=ml^2)

Consequently

MgH=2ml^2W^2
12. Formulate the law of conservation of total mechanical energy in relation to a body rotating around a fixed axis.
The angular momentum of a rotating body is directly proportional to the speed of rotation of the body, its mass and linear extent. The higher any of these values, the higher the angular momentum.

In mathematical representation, the angular momentum L a body rotating at an angular velocity ω , is equal to L = Iω, where the value I called moment of inertia

Angular moment of a rotating body

where is the body weight; - speed; is the radius of the orbit along which the body moves; - moment of inertia; is the angular velocity of the rotating body.

Law of conservation of angular momentum:

– for rotational movement

13. What expression determines the work of the moment of forces

= TORQUE * ANGLE

In the SI system, work is measured in Joules, moment of force in Newton * meter, and ANGLE in radians

Usually known is the angular velocity in radians per second and the duration of the TORQUE.

Then the WORK done by the TORQUE of force is calculated as:

= MOMENT OF POWER * *

14. Get a formula that determines the power developed by the moment of forces.
If a force performs an action at any distance, then it performs mechanical work. Also, if a moment of force performs an action through an angular distance, it does work.

= TORQUE_FORCE * ANGULAR_SPEED

In the SI system, power is measured in watts, torque in Newton meters, and ANGULAR VELOCITY in radians per second.

15. Get a formula that determines the power developed by the moment of forces.

Forces and moments of forces act on the links of the mechanism, developing the corresponding powers. Thus, the power of all given forces consists of two parts:
,
where N R- the power developed by the forces applied at various points of the links performing translational or complex plane motion; N M - the power developed by the moments of forces applied to the rotating links.

Then , PowerN M calculated by the formula:
,
whereM k - moment acting onk -e rotating links; w k are the angular velocities of these links.
16. What is the kinetic energy of the rolling body?

During the rotational motion of a rolling body, each point participates in 2 movements - translational and rotational.

17. in my opinion, the moment of force will increase / decrease by 2 times (direct dependence)

the moment of inertia is the same
18. moment of force will increase / decrease by 2 times

moment of inertia increased / decreased by 4 times

22. Why is laboratory setup #4 called Oberbeck's PENDULUM?

A load hangs on the back of the thread. Under the influence of gravity, this weight pulls the block. And because of this, the pendulum begins to spin. When the thread ends, stretches, and the load falls, the pendulum continues to spin due to inertia until it stops. If the thread is fixed, then when it ends and is pulled, the pendulum continues to rotate by inertia, so the thread begins to wind again, and the load, accordingly, rises. And then it will stop and start going down again. And in this process of raising and lowering lies the meaning of the pendulum.
23. How does taking into account the forces of friction affect the result of measuring the moment of inertia of the Oberbeck pendulum? In which case is the measured value of the moment of inertia of the Oberbeck pendulum greater (with or without frictional forces)? Justify the answer.

If the friction force is taken into account, then the equation looks like this: .

That is, (if we derive from this formula I) the friction force helps to reduce the moment of inertia of the rigid body. Consequently, the measured value of the moment of inertia of the pendulum without taking into account the forces of friction will be greater than with their allowance.

24. What forces act on the falling weight of an Oberbeck pendulum? What are they equal to?

Per cargo valid his strength gravity ([ mg ]=1 Newton) and strengththread tension ([ T ]=1 Newton).

The force of gravity acts on the load in the downward direction Fgr = mg,

where m is the mass of the load, and g is the acceleration due to gravity (9.8 m/(s^2).

Since the load is motionless, and other than the force of gravity and the tension of the thread, other forces do not act on it, then according to Newton's second law T = Ftight = mg, where T is the tension force of the thread.

If the load at the same time moves uniformly, that is, without acceleration, then T is also equal to mg according to Newton's first law.

If a load with a mass m moves down with an acceleration a.

Then, according to Newton's second law, Fstrand-T = mg-T = ma. Thus, T = mg-a.
25. A person stands in the center of a rotating platform (carousel). How will the rotation speed of the platform change if a person moves to the edge of the platform.

The (instantaneous) velocity vector of any point of an (absolutely) rigid body rotating at an angular velocity is given by:

where is the radius vector to the given point from the origin located on the axis of rotation of the body, and square brackets denote the vector product.

Linear velocity (coinciding with the module of the velocity vector) of a point at a certain distance (radius)from the axis of rotation can be calculated as follows:

Therefore, the greater the distance, the greater the speed. This means the carousel will spin faster.
26. The hoop and the solid cylinder have the same masses and radii. Determine their kinetic energies if they roll at the same speed.

Kinetic energyrotary motion- energy body associated with its rotation.

For absolutely rigid bodythe total kinetic energy can be written as the sum of the kinetic energy of translational and rotational motion:

Axial moments of inertia

Cylinder

Speed \u003d R * ω

In the photo, the W formulas are the T formulas. We found them to. Energy and the ratio of energies.
27. What is the moment of force if the direction of the force is: a / perpendicular to the axis of rotation, b / parallel to the axis of rotation, c / passes through the axis of rotation.
A. M = +/- Fh

B. The moment of the force about the axis is zero if the force is parallel to the axis. In this case, the projection of the force on a plane perpendicular to the axis is equal to zero.

B. The moment of force about an axis is zero if the line of action of the force intersects this axis. In this case, the line of action of the force on a plane perpendicular to the axis passes through the point of intersection of the axis with the plane and, therefore, the arm of the force relative to the point O is equal to zero.

28. ???

29. What is the center of gravity of a rigid body?

center of gravityof a rigid body is a point invariably associated with this bodyFROM, through which the line of action of the resultant gravity of a given body passes, for any position of the body in space.

30. In what two ways can the moment of force that rotates the Oberebek's pendulum be changed?

31. In what two ways can the moment of force be changed without changing the point of application of the force?

Change force value or radius

32. What formula can be used to theoretically calculate the total moment of inertia of the weights on the spokes of the Oberbeck pendulum? Explain the quantities included in it.

− weighti-th material point

- distance of a material point to the considered axis

33. Specify the direction of the vector of angular acceleration of a rotating body with a fixed axis of rotation relative to the vector of angular velocity.

When the body rotates around a fixed axis, the angular acceleration vector is directed along the rotation axis towards the vector of the elementary increment of the angular velocity. With accelerated movement, the vectorEco-directed to the vectorW, when slowed down, it is opposite to it.

Eis the angular acceleration vector

Wis the angular velocity vector

34. Using the measurement data, calculate the work of the friction forces during the rotation of the Oberbeck pendulum at the moment of impact of the falling weight on the floor.
35. Using the measurement data, calculate the kinetic energy of rotation of the Oberbeck pendulum at the moment the falling weight hits the floor.

E vr - kinetic energy of a rotating flywheel with a load.

I- flywheel moment of inertia (together with weights);  - angular speed of rotation of the flywheel at the moment of impact of the weight with the floor.

36. Using the measurement data, calculate the potential energy of the falling weight of the Oberbeck pendulum before the system begins to move.

m is the mass of the load, h is its height above the floor

37. What is called a "pair of forces", write a formula, determine the moment of the "pair of forces", where is it directed?

A pair of forces is a system of two equal in magnitude, opposite in direction and not lying on the same straight line of forces. A pair of forces exerts a rotating action, which can be estimated by the moment of the pair:

M(F 1 ,F 2)=F 1 h=F 2 h

where h is the arm of the pair, i.e. the distance between the lines of action of the couple's forces.

The moment of the pair of forces M is perpendicular to the plane of action of the pair ( the plane in which the vectors of the pair of forces are located) and directed according to the right screw rule. The vector moment of a pair of forces can be applied at any point in space, i.e. is an free vector.

38. What types of energy does the potential energy of a falling weight transform into when the Oberbeck pendulum rotates?

The potential energy of the falling weight is converted into the kinetic energy of the translational motion of this weight and the kinetic energy of the rotational motion of the pendulum.

39. What types of energy does the kinetic energy of an Oberbeck pendulum transform into when it rotates?

Potential?

40. Draw the forces acting on the falling weight, what are they equal to? What is the nature of the movement of the falling weight?

T - thread tension, mg - gravity

A falling weight moves with uniform acceleration.

Date: __________ Deputy Director for OIA: ___________

Topic; Newton's second law for rotational motion

Target:

Educational: determine and write down in mathematical form Newton's second law; explain the relationship between the quantities included in the formulas of this law;

Developing: develop logical thinking, the ability to explain the manifestations of Newton's second law in nature;

Educational : to form interest in the study of physics, to cultivate diligence, responsibility.

Type of lesson: learning new material.

Demonstrations: the dependence of the acceleration of a body on the force acting on it.

Equipment: trolley with light wheels, rotating disk, set of weights, spring, block, bar.

DURING THE CLASSES

Organizing time

Updating the basic knowledge of students

Formula chain (reproduce formulas):

II. Motivation of educational activity of students

Teacher. With the help of Newton's laws, one can not only explain the observed mechanical phenomena, but also predict their course. Recall that the direct main task of mechanics is to find the position and speed of a body at any moment of time, if its position and speed at the initial moment of time and the forces that act on it are known. This problem is solved with the help of Newton's second law, which we will study today.

III. Learning new material

1. The dependence of the acceleration of a body on the force acting on it

A more inert body has a large mass, a less inert body has a smaller one:

2. Newton's second law

Newton's second law of dynamics establishes a connection between kinematic and dynamic quantities. Most often, it is formulated as follows: the acceleration that a body receives is directly proportional to the mass of the body and has the same direction as the force:

where - acceleration, - resultant of forces acting on the body, N; m - body weight, kg.

If we determine the force from this expression, then we obtain the second law of dynamics in the following formulation: the force acting on the body is equal to the product of the body's mass and the acceleration provided by this force.

Newton formulated the second law of dynamics somewhat differently, using the concept of momentum (body momentum). Impulse - the product of body mass and its speed (the same as the amount of motion) - one of the measures of mechanical movement: Impulse (momentum) is a vector quantity. Since the acceleration

Newton formulated his law as follows: the change in the momentum of a body is proportional to the acting force and occurs in the direction of the straight line along which this force acts.

It is worth considering another of the formulations of the second law of dynamics. In physics, a vector quantity is widely used, which is called the impulse of a force - this is the product of the force and the time of its action: Using this, we get . The change in momentum of a body is equal to the momentum of the force acting on it.

Newton's second law of dynamics summarized an extremely important fact: the action of forces does not cause actual motion, but only changes it; force causes a change in speed, i.e. acceleration, not speed itself. The direction of the force coincides with the direction of the velocity only in the partial case of rectilinear evenly accelerated (Δ 0) motion. For example, during the movement of a body thrown horizontally, the force of gravity is directed downward, and the velocity forms a certain angle with the force, which changes during the flight of the body. And in the case of uniform motion of the body in a circle, the force is always directed perpendicular to the speed of the body.

The SI unit of force is determined based on Newton's second law. The unit of force is called [H] and is defined as follows: a force of 1 newton imparts an acceleration of 1 m/s2 to a body of mass 1 kg. In this way,

Application examples of Newton's second law

As an example of the application of Newton's second law, one can consider, in particular, the measurement of body mass by weighing. An example of the manifestation of Newton's second law in nature can be a force that acts on our planet from the Sun, etc.

Limits of application of Newton's second law:

1) the reference system must be inertial;

2) the speed of the body must be much less than the speed of light (for speeds close to the speed of light, Newton's second law is used in impulsive form: ).

IV. Fixing the material

Problem solving

1. A body with a mass of 500 g is simultaneously affected by two forces 12 N and 4 N, directed in the opposite direction along one straight line. Determine the modulus and direction of acceleration.

Given: m = 500 g = 0.5 kg, F1 = 12 N, F2 = 4 N.

Find: a - ?

According to Newton's second law: , where Let's draw the axis Ox, then the projection F = F1 - F2. In this way,

Answer: 16 m/s2, the acceleration is in the direction of the greater force.

2. The coordinate of the body changes according to the law x = 20 + 5t + 0.5t2 under the action of a force of 100 N. Find the mass of the body.

Given: x = 20 + 5t + 0.5t2, F = 100H

Find: m - ?

Under the action of a force, the body moves with equal acceleration. Therefore, its coordinate changes according to the law:

According to Newton's second law:

Answer: 100 kg.

3. A body with a mass of 1.2 kg acquired a speed of 12 m/s at a distance of 2.4 m under the action of a force of 16 N. Find the initial velocity of the body.

Given: = 12 m/s, s = 2.4m, F = 16H, m = 1.2 kg

Find: 0 - ?

Under the action of a force, the body acquires acceleration according to Newton's second law:

For evenly accelerated movement:

From (2) we express the time t:

and substitute for t in (1):

Substitute the expression for acceleration:

Answer: 8.9 m/s.

V. Lesson summary

Frontal conversation for questions

1. How are such physical quantities as acceleration, force and mass of a body related?

2. Or can it be stated by the formula that the force acting on a body depends on its mass and acceleration?

3. What is the momentum of the body (momentum)?

4. What is the impulse of force?

5. What formulations of Newton's second law do you know?

6. What important conclusion can be drawn from Newton's second law?

VI. Homework

Work through the relevant section of the textbook.

Solve problems:

1. Find the acceleration module of a body with a mass of 5 kg under the action of four forces applied to it, if:

a) F1 = F3 = F4 = 20 H, F2 = 16 H;

b) F1 = F4 = 20 H, F2 = 16 H, F3 = 17 H.

2. A body with a mass of 2 kg, moving in a straight line, changed its speed from 1 m/s to 2 m/s in 4 s.

a) What is the acceleration of the body?

b) What force acted on the body in the direction of its motion?

c) How has the momentum of the body (momentum) changed over the time considered?

d) What is the impulse of the force acting on the body?

e) What is the distance traveled by the body during the considered time of motion?