Moment of inertia of a body about an axis. Basic laws and formulas in theoretical mechanics

Theoretical mechanics is a section of mechanics that sets out the basic laws of mechanical motion and mechanical interaction of material bodies.

Theoretical mechanics is a science that studies the movement of bodies over time (mechanical movements). It serves as the basis for other branches of mechanics (theory of elasticity, strength of materials, theory of plasticity, theory of mechanisms and machines, hydroaerodynamics) and many technical disciplines.

Mechanical movement- this is a change over time in the relative position in space of material bodies.

Mechanical interaction- this is an interaction as a result of which the mechanical movement changes or the relative position of body parts changes.

Rigid body statics

Statics is a section of theoretical mechanics that deals with problems of equilibrium of solid bodies and the transformation of one system of forces into another, equivalent to it.

    Basic concepts and laws of statics
  • Absolutely rigid body(solid body, body) is a material body, the distance between any points in which does not change.
  • Material point is a body whose dimensions, according to the conditions of the problem, can be neglected.
  • Free body- this is a body on the movement of which no restrictions are imposed.
  • Unfree (bound) body is a body whose movement is subject to restrictions.
  • Connections– these are bodies that prevent the movement of the object in question (a body or a system of bodies).
  • Communication reaction is a force that characterizes the action of a bond on a solid body. If we consider the force with which a solid body acts on a bond to be an action, then the reaction of the bond is a reaction. In this case, the force - action is applied to the connection, and the reaction of the connection is applied to the solid body.
  • Mechanical system is a collection of interconnected bodies or material points.
  • Solid can be considered as a mechanical system, the positions and distances between points of which do not change.
  • Force is a vector quantity that characterizes the mechanical action of one material body on another.
    Force as a vector is characterized by the point of application, direction of action and absolute value. The unit of force modulus is Newton.
  • Line of action of force is a straight line along which the force vector is directed.
  • Focused Power– force applied at one point.
  • Distributed forces (distributed load)- these are forces acting on all points of the volume, surface or length of a body.
    The distributed load is specified by the force acting per unit volume (surface, length).
    The dimension of the distributed load is N/m 3 (N/m 2, N/m).
  • External force is a force acting from a body that does not belong to the mechanical system under consideration.
  • Inner strength is a force acting on a material point of a mechanical system from another material point belonging to the system under consideration.
  • Force system is a set of forces acting on a mechanical system.
  • Flat force system is a system of forces whose lines of action lie in the same plane.
  • Spatial system of forces is a system of forces whose lines of action do not lie in the same plane.
  • System of converging forces is a system of forces whose lines of action intersect at one point.
  • Arbitrary system of forces is a system of forces whose lines of action do not intersect at one point.
  • Equivalent force systems- these are systems of forces, the replacement of which one with another does not change the mechanical state of the body.
    Accepted designation: .
  • Equilibrium- this is a state in which a body, under the action of forces, remains motionless or moves uniformly in a straight line.
  • Balanced system of forces- this is a system of forces that, when applied to a free solid body, does not change its mechanical state (does not throw it out of balance).
    .
  • Resultant force is a force whose action on a body is equivalent to the action of a system of forces.
    .
  • Moment of power is a quantity characterizing the rotating ability of a force.
  • Couple of forces is a system of two parallel forces of equal magnitude and oppositely directed.
    Accepted designation: .
    Under the influence of a pair of forces, the body will perform a rotational movement.
  • Projection of force on the axis- this is a segment enclosed between perpendiculars drawn from the beginning and end of the force vector to this axis.
    The projection is positive if the direction of the segment coincides with the positive direction of the axis.
  • Projection of force onto a plane is a vector on a plane, enclosed between perpendiculars drawn from the beginning and end of the force vector to this plane.
  • Law 1 (law of inertia). An isolated material point is at rest or moves uniformly and rectilinearly.
    The uniform and rectilinear motion of a material point is motion by inertia. The state of equilibrium of a material point and a rigid body is understood not only as a state of rest, but also as motion by inertia. For a rigid body, there are various types of motion by inertia, for example, uniform rotation of a rigid body around a fixed axis.
  • Law 2. A rigid body is in equilibrium under the action of two forces only if these forces are equal in magnitude and directed in opposite directions along a common line of action.
    These two forces are called balancing.
    In general, forces are called balanced if the solid body to which these forces are applied is at rest.
  • Law 3. Without disturbing the state (the word “state” here means the state of motion or rest) of a rigid body, one can add and reject balancing forces.
    Consequence. Without disturbing the state of the solid body, the force can be transferred along its line of action to any point of the body.
    Two systems of forces are called equivalent if one of them can be replaced by the other without disturbing the state of the solid body.
  • Law 4. The resultant of two forces applied at one point, applied at the same point, is equal in magnitude to the diagonal of a parallelogram constructed on these forces, and is directed along this
    diagonals.
    The absolute value of the resultant is:
  • Law 5 (law of equality of action and reaction). The forces with which two bodies act on each other are equal in magnitude and directed in opposite directions along the same straight line.
    It should be kept in mind that action- force applied to the body B, And opposition- force applied to the body A, are not balanced, since they are applied to different bodies.
  • Law 6 (law of solidification). The equilibrium of a non-solid body is not disturbed when it solidifies.
    It should not be forgotten that the equilibrium conditions, which are necessary and sufficient for a solid body, are necessary but insufficient for the corresponding non-solid body.
  • Law 7 (law of emancipation from ties). A non-free solid body can be considered as free if it is mentally freed from bonds, replacing the action of the bonds with the corresponding reactions of the bonds.
    Connections and their reactions
  • Smooth surface limits movement normal to the support surface. The reaction is directed perpendicular to the surface.
  • Articulated movable support limits the movement of the body normal to the reference plane. The reaction is directed normal to the support surface.
  • Articulated fixed support counteracts any movement in a plane perpendicular to the axis of rotation.
  • Articulated weightless rod counteracts the movement of the body along the line of the rod. The reaction will be directed along the line of the rod.
  • Blind seal counteracts any movement and rotation in the plane. Its action can be replaced by a force represented in the form of two components and a pair of forces with a moment.

Kinematics

Kinematics- a section of theoretical mechanics that examines the general geometric properties of mechanical motion as a process occurring in space and time. Moving objects are considered as geometric points or geometric bodies.

    Basic concepts of kinematics
  • Law of motion of a point (body)– this is the dependence of the position of a point (body) in space on time.
  • Point trajectory– this is the geometric location of a point in space during its movement.
  • Speed ​​of a point (body)– this is a characteristic of the change in time of the position of a point (body) in space.
  • Acceleration of a point (body)– this is a characteristic of the change in time of the speed of a point (body).
    Determination of kinematic characteristics of a point
  • Point trajectory
    In a vector reference system, the trajectory is described by the expression: .
    In the coordinate reference system, the trajectory is determined by the law of motion of the point and is described by the expressions z = f(x,y)- in space, or y = f(x)- in a plane.
    In a natural reference system, the trajectory is specified in advance.
  • Determining the speed of a point in a vector coordinate system
    When specifying the movement of a point in a vector coordinate system, the ratio of movement to a time interval is called the average value of speed over this time interval: .
    Taking the time interval to be an infinitesimal value, we obtain the speed value at a given time (instantaneous speed value): .
    The average velocity vector is directed along the vector in the direction of the point’s movement, the instantaneous velocity vector is directed tangentially to the trajectory in the direction of the point’s movement.
    Conclusion: the speed of a point is a vector quantity equal to the time derivative of the law of motion.
    Derivative property: the derivative of any quantity with respect to time determines the rate of change of this quantity.
  • Determining the speed of a point in a coordinate reference system
    Rate of change of point coordinates:
    .
    The modulus of the total velocity of a point with a rectangular coordinate system will be equal to:
    .
    The direction of the velocity vector is determined by the cosines of the direction angles:
    ,
    where are the angles between the velocity vector and the coordinate axes.
  • Determining the speed of a point in a natural reference system
    The speed of a point in the natural reference system is defined as the derivative of the law of motion of the point: .
    According to previous conclusions, the velocity vector is directed tangentially to the trajectory in the direction of the point’s movement and in the axes is determined by only one projection.
    Rigid body kinematics
  • In the kinematics of rigid bodies, two main problems are solved:
    1) setting the movement and determining the kinematic characteristics of the body as a whole;
    2) determination of kinematic characteristics of body points.
  • Translational motion of a rigid body
    Translational motion is a motion in which a straight line drawn through two points of a body remains parallel to its original position.
    Theorem: during translational motion, all points of the body move along identical trajectories and at each moment of time have the same magnitude and direction of speed and acceleration.
    Conclusion: the translational motion of a rigid body is determined by the movement of any of its points, and therefore, the task and study of its motion is reduced to the kinematics of the point.
  • Rotational motion of a rigid body around a fixed axis
    Rotational motion of a rigid body around a fixed axis is the motion of a rigid body in which two points belonging to the body remain motionless during the entire time of movement.
    The position of the body is determined by the angle of rotation. The unit of measurement for angle is radian. (A radian is the central angle of a circle, the arc length of which is equal to the radius; the total angle of the circle contains 2π radian.)
    The law of rotational motion of a body around a fixed axis.
    We determine the angular velocity and angular acceleration of the body using the differentiation method:
    — angular velocity, rad/s;
    — angular acceleration, rad/s².
    If you dissect the body with a plane perpendicular to the axis, select a point on the axis of rotation WITH and an arbitrary point M, then point M will describe around a point WITH circle radius R. During dt there is an elementary rotation through an angle , and the point M will move along the trajectory a distance .
    Linear speed module:
    .
    Point acceleration M with a known trajectory, it is determined by its components:
    ,
    Where .
    As a result, we get the formulas
    tangential acceleration: ;
    normal acceleration: .

Dynamics

Dynamics is a section of theoretical mechanics in which the mechanical movements of material bodies are studied depending on the causes that cause them.

    Basic concepts of dynamics
  • Inertia- this is the property of material bodies to maintain a state of rest or uniform rectilinear motion until external forces change this state.
  • Weight is a quantitative measure of the inertia of a body. The unit of mass is kilogram (kg).
  • Material point- this is a body with mass, the dimensions of which are neglected when solving this problem.
  • Center of mass of a mechanical system- a geometric point whose coordinates are determined by the formulas:

    Where m k , x k , y k , z k— mass and coordinates k-that point of the mechanical system, m— mass of the system.
    In a uniform field of gravity, the position of the center of mass coincides with the position of the center of gravity.
  • Moment of inertia of a material body relative to an axis is a quantitative measure of inertia during rotational motion.
    The moment of inertia of a material point relative to the axis is equal to the product of the mass of the point by the square of the distance of the point from the axis:
    .
    The moment of inertia of the system (body) relative to the axis is equal to the arithmetic sum of the moments of inertia of all points:
  • Inertia force of a material point is a vector quantity equal in modulus to the product of the mass of a point and the acceleration modulus and directed opposite to the acceleration vector:
  • The force of inertia of a material body is a vector quantity equal in modulus to the product of the body mass and the modulus of acceleration of the center of mass of the body and directed opposite to the acceleration vector of the center of mass: ,
    where is the acceleration of the center of mass of the body.
  • Elementary impulse of force is a vector quantity equal to the product of the force vector and an infinitesimal period of time dt:
    .
    The total force impulse for Δt is equal to the integral of the elementary impulses:
    .
  • Elementary work of force is a scalar quantity dA, equal to the scalar proi

Let there be a solid body. Let's choose some straight line OO (Fig. 6.1), which we will call an axis (the straight line OO can be outside the body). Let us divide the body into elementary sections (material points) with masses
located at a distance from the axis
respectively.

The moment of inertia of a material point relative to an axis (OO) is the product of the mass of a material point by the square of its distance to this axis:


. (6.1)

The moment of inertia (MI) of a body relative to an axis (OO) is the sum of the products of the masses of elementary sections of the body by the square of their distance to the axis:

. (6.2)

As you can see, the moment of inertia of a body is an additive quantity - the moment of inertia of the entire body relative to a certain axis is equal to the sum of the moments of inertia of its individual parts relative to the same axis.

In this case

.

The moment of inertia is measured in kgm 2. Because

, (6.3)

where  – density of the substance,
- volume i- th section, then

,

or, moving to infinitesimal elements,

. (6.4)

Formula (6.4) is convenient to use to calculate the MI of homogeneous bodies of regular shape relative to the axis of symmetry passing through the center of mass of the body. For example, for the MI of a cylinder relative to an axis passing through the center of mass parallel to the generatrix, this formula gives

,

Where T- weight; R- radius of the cylinder.

Steiner's theorem provides great assistance in calculating the MI of bodies relative to certain axes: MI of bodies I relative to any axis is equal to the sum of the MI of this body I c relative to an axis passing through the center of mass of the body and parallel to the given one, and the product of the body mass by the square of the distance d between the indicated axes:

. (6.5)

Moment of force about the axis

Let the force act on the body F. Let us assume for simplicity that the force F lies in a plane perpendicular to some straight line OO (Fig. 6.2, A), which we will call the axis (for example, this is the axis of rotation of the body). In Fig. 6.2, A A- point of application of force F,
- the point of intersection of the axis with the plane in which the force lies; r- radius vector defining the position of the point A relative to the point ABOUT"; O"B = b - shoulder of strength. The force arm relative to the axis is the smallest distance from the axis to the straight line on which the force vector lies F(the length of the perpendicular drawn from the point to this line).

The moment of force relative to the axis is a vector quantity defined by the equality

. (6.6)

The modulus of this vector is . Sometimes, therefore, they say that the moment of a force about an axis is the product of the force and its arm.

If strength F is directed arbitrarily, then it can be decomposed into two components; And (Fig.6.2, b), i.e.
+, Where - component directed parallel to the OO axis, and lies in a plane perpendicular to the axis. In this case, under the moment of force F relative to the OO axis understand the vector

. (6.7)

In accordance with expressions (6.6) and (6.7), the vector M directed along the axis (see Fig. 6.2, A,b).

Momentum of a body relative to the axis of rotation

P Let the body rotate around a certain axis OO with angular velocity
. Let's mentally break this body down into elementary sections with masses
, which are located from the axis, respectively, at distances
and rotate in circles, having linear speeds
It is known that the value is equal
- there is an impulse i-plot. moment of impulse i-section (material point) relative to the axis of rotation is called a vector (more precisely, a pseudovector)

, (6.8)

Where r i– radius vector defining the position i- area relative to the axis.

The angular momentum of the entire body relative to the axis of rotation is called the vector

(6.9)

whose module
.

In accordance with expressions (6.8) and (6.9), the vectors
And directed along the axis of rotation (Fig. 6.3). It is easy to show that the angular momentum of a body L relative to the axis of rotation and moment of inertia I of this body relative to the same axis are related by the relation

. (6.10)

The moment of inertia of a body (system) relative to a given axis Oz (or axial moment of inertia) is a scalar quantity that is different from the sum of the products of the masses of all points of the body (system) by the squares of their distances from this axis:

From the definition it follows that the moment of inertia of a body (or system) relative to any axis is a positive quantity and not equal to zero.

In the future, it will be shown that the axial moment of inertia plays the same role during rotational motion of a body as mass does during translational motion, i.e., that the axial moment of inertia is a measure of the inertia of a body during rotational motion.

According to formula (2), the moment of inertia of a body is equal to the sum of the moments of inertia of all its parts relative to the same axis. For one material point located at a distance h from the axis, . The unit of measurement of the moment of inertia in SI will be 1 kg (in the MKGSS system -).

To calculate the axial moments of inertia, the distances of points from the axes can be expressed through the coordinates of these points (for example, the square of the distance from the Ox axis will be, etc.).

Then the moments of inertia about the axes will be determined by the formulas:

Often during calculations the concept of radius of gyration is used. The radius of inertia of a body relative to an axis is a linear quantity determined by the equality

where M is body mass. From the definition it follows that the radius of inertia is geometrically equal to the distance from the axis of the point at which the mass of the entire body must be concentrated so that the moment of inertia of this one point is equal to the moment of inertia of the entire body.

Knowing the radius of inertia, you can use formula (4) to find the moment of inertia of the body and vice versa.

Formulas (2) and (3) are valid both for a rigid body and for any system of material points. In the case of a solid body, breaking it into elementary parts, we find that in the limit the sum in equality (2) will turn into an integral. As a result, taking into account that where is the density and V is the volume, we obtain

The integral here extends to the entire volume V of the body, and the density and distance h depend on the coordinates of the points of the body. Similarly, formulas (3) for solid bodies take the form

Formulas (5) and (5) are convenient to use when calculating the moments of inertia of homogeneous bodies of regular shape. In this case, the density will be constant and will fall outside the integral sign.

Let us find the moments of inertia of some homogeneous bodies.

1. A thin homogeneous rod of length l and mass M. Let us calculate its moment of inertia relative to the axis perpendicular to the rod and passing through its end A (Fig. 275). Let us direct the coordinate axis along AB. Then for any elementary segment of length d the value is , and the mass is , where is the mass of a unit length of the rod. As a result, formula (5) gives

Replacing here with its value, we finally find

2. A thin round homogeneous ring of radius R and mass M. Let us find its moment of inertia relative to the axis perpendicular to the plane of the ring and passing through its center C (Fig. 276).

Since all points of the ring are located at a distance from the axis, formula (2) gives

Therefore, for the ring

Obviously, the same result will be obtained for the moment of inertia of a thin cylindrical shell of mass M and radius R relative to its axis.

3. A round homogeneous plate or cylinder of radius R and mass M. Let us calculate the moment of inertia of the round plate relative to the axis perpendicular to the plate and passing through its center (see Fig. 276). To do this, we select an elementary ring with radius and width (Fig. 277, a). The area of ​​this ring is , and the mass is where is the mass per unit area of ​​the plate. Then, according to formula (7) for the selected elementary ring there will be and for the entire plate

In mechanics, a rigid body is understood as a system of material points, the distance between any two points of which remains unchanged during motion. Therefore, all the results obtained in the previous topics (“Dynamics of a material point”, “Law of conservation of momentum”, “Law of conservation of energy” and “Law of conservation of angular momentum”) for a system of material points are also applicable to a solid body.

Moment of inertia of a rigid body

The moment of inertia is a quantity that depends on the distribution of masses in a body and is, along with mass, a measure of the inertia of a body during non-translational motion. When a rigid body rotates around a fixed axis, the moment of inertia of the body relative to this axis is determined by the expression

Where - elementary body masses; - their distances from the axis of rotation.

The moment of inertia of a body relative to any axis can be found by calculation. If the matter in the body is distributed continuously, then calculating the moment of inertia is reduced to calculating the integral

, (1)

Where
– mass of a body element located at a distance from the axis of interest to us. Integration must be carried out over the entire volume of the body.

Analytical calculation of such integrals is possible only in the simplest cases of bodies of regular geometric shape.

If the moment of inertia of a body about any axis is known, you can find the moment of inertia about any other axis parallel to this one. Using Steiner's theorem, according to which the moment of inertia of a body relative to an arbitrary axis is equal to the sum of the moment of inertia of the body relative to the axis passing through the center of mass of the body and parallel to a given axis, and the product of body mass T per square of the distance between the axes :

(2)

Calculation of the moment of inertia of a body relative to an axis can often be simplified by first calculating moment of inertia about a point. The moment of inertia of a body relative to a point itself does not play any role in dynamics. It is a purely auxiliary concept that serves to simplify calculations.

Let us consider some point of a rigid body with mass and with coordinates
relative to the rectangular coordinate system (Fig. 1). Squares of its distances to the coordinate axes
are equal respectively


and the moments of inertia about the same axes



(3)

Adding these equalities and summing over the entire volume of the body

(5)

Where
– moment of inertia of the body relative to the point.

From this expression we can obtain the relationship between the moments of inertia of a flat body relative to the axes
. Let the mass of a flat body be concentrated in the plane
those. coordinate any point of such a body is equal to zero, then from

equations (3) and (4) it follows that

(6)

Rotation of a rigid body around a fixed axis

Consider a solid body of mass , rotating around a fixed axis with angular velocity . In order to obtain an equation describing this movement, we apply the equation of moments about the axis obtained in the section “Law of conservation of angular momentum”

, (7)

Recall that in this equation And
– angular momentum and moment of force about the axis around which the rigid body rotates.

The angular momentum of a certain point of a body of mass
rotating in a circle of radius with speed , is equal

Summing over the entire volume of the body, taking into account that
we get

Thus, the angular momentum of a rigid body rotating around a fixed axis is equal to the product of the moment of inertia of the body relative to this axis and its angular velocity.

Substituting the resulting expression into (7), we obtain the equation of dynamics of a rigid body rotating around a fixed axis,

or
(8)

Where – angular acceleration of the body.

Let's find the kinetic energy of a rotating body. To do this, we sum up the kinetic energies of its individual parts over the entire volume of the body.

(9)

Knowing the dependence of the moment of forces acting on the body on the angle of rotation, one can find the work of these forces when the body rotates through a finite angle

.

MOMENT OF INERTIA I of a body relative to a point, axis or plane is called the sum of the products of the mass of points of the body m i by the squares of their distances r i to the point, axis or plane:

The moment of inertia of a body about an axis is a measure of the inertia of a body in rotational motion around that axis.

The moment of inertia of a body can also be expressed in terms of the mass M of the body and its radius of gyration r:

MOMENTS OF INERTIA RELATIVE TO AXES, PLANES AND ORIGIN OF CARTESIAN COORDINATES.

Moment of inertia about the origin (polar moment of inertia):

RELATIONSHIP BETWEEN AXIAL, PLANE AND POLAR MOMENTS OF INERTIA:

The values ​​of the axial moments of inertia of some geometric bodies are given in Table. 1.

Table 1. Moment of inertia of some bodies
Figure or body

At c→0 a rectangular plate is obtained

CHANGE IN MOMENTS OF INERTIA WHEN CHANGING AXES

Moment of inertia I u 1 relative to the u 1 axis parallel to the given u axis (Fig. 1):

where I u is the moment of inertia of the body relative to the u axis; l(l 1) - distance from the u axis (from the u 1 axis) to the u c axis parallel to them, passing through the center of mass of the body; a is the distance between the u and u 1 axes.

Picture 1.

If the u axis is central (l=0), then

that is, for any group of parallel axes, the moment of inertia about the central axis is the smallest.

Moment of inertia I u relative to the u axis making angles α, β, γ with the Cartesian coordinate axes x, y, z (Fig. 2):

Figure 2.

The x, y, z axes are main if

Moment of inertia relative to the u axis making angles α, β, γ with the main axes of inertia x, y, z:

CHANGE IN CENTRIFUGAL MOMENTS OF INERTIA DURING PARALLEL TRANSFER OF AXES:

where is the centrifugal moment of inertia relative to the central axes x c, y c, parallel to the x, y axes; M - body weight; x с, y с - coordinates of the center of mass in the x, y axes system.

CHANGE IN CENTRIFUGAL MOMENT OF INERTIA WHEN AXES x, y ROTATE AROUND AXIS z BY ANGLE α TO POSITION x 1 y 1(Fig. 3):

Figure 3.

DETERMINATION OF THE POSITION OF THE MAIN AXES OF INERTIA. The axis of material symmetry of the body is the main axis of inertia of the body.

If the xOz plane is the plane of material symmetry of the body, then any of the y axes is the main axis of inertia of the body.

If the position of one of the main axes z main is known, then the position of the other two axes x main and y main is determined by rotating the x and y axes around the z main axis by an angle φ (Fig. 3):

ELLIPSOID AND PARALLELEPIPED OF INERTIA. An ellipsoid of inertia is an ellipsoid whose symmetry axes coincide with the main central axes of the body x main, y main, z main, and the semi-axes a x, a y, a z are equal, respectively:

where r уО z, r x Oz, r xOy are the radii of inertia of the body relative to the main planes of inertia.

A parallelepiped of inertia is a parallelepiped that is described around the ellipsoid of inertia and has common axes of symmetry with it (Fig. 4).

Figure 4.

REDUCTION (REPLACEMENT TO SIMPLIFY CALCULATIONS) OF A SOLID BODY WITH CONCENTRATED MASSES. When calculating axial, planar, centrifugal and polar moments of inertia, a body of mass M can be reduced by eight concentrated masses M/8 located at the vertices of the parallelepiped of inertia. Moments of inertia relative to any axes, planes, poles are calculated from the coordinates of the vertices of the parallelepiped of inertia x i, y i, z i (i=1, 2, ..., 8) using the formulas:

EXPERIMENTAL DETERMINATION OF MOMENTS OF INERTIA

1. Determination of the moments of inertia of bodies of rotation using the differential equation of rotation - see formulas ("Rotational motion of a rigid body").

The body under study is fixed on the horizontal axis x, coinciding with its axis of symmetry, and is brought into rotation around it using a load P attached to a flexible thread wound onto the body under study (Fig. 5), while the time t of lowering the load to a height h is measured . To eliminate the influence of friction at the points of fastening of the body on the x axis, the experiment is carried out several times at different values ​​of the weight of the load P.

Figure 5.

In two experiments with loads P 1 and P 2

2. Experimental determination of the moments of inertia of bodies by studying the oscillations of a physical pendulum (see 2.8.3) .

The body under study is fixed on the horizontal axis x (non-central) and the period of small oscillations around this axis T is measured. The moment of inertia about the x axis is determined by the formula

where P is body weight; l 0 - distance from the axis of rotation to the center of mass C of the body.