Calculation of the moment of inertia of a pendulum. Determination of the moment of inertia of a physical pendulum Determination of the moments of inertia of physical pendulums of various shapes

When solving equations of rotational or oscillatory (oscillating) motion, it is necessary to know the moment of inertia of the system under consideration. This article is devoted to the study of various types of pendulums and the moment of inertia with which they are characterized.

The concept of a pendulum. Species

Before giving the definition of the moment of inertia of a pendulum, it is necessary to consider what this device is. In physics, it is understood as absolutely any system that can oscillate or rotate around a certain point or axis under the influence of a gravitational field, that is, the force of gravity. This definition assumes that the pendulum must necessarily have a finite mass, while the center of mass of the system should not be at the point through which the axis of rotation passes.

There are different types of pendulums. In this article we will consider only 3 of them:

mathematical, or simple;
physical (using the example of a homogeneous rod);
Oberbeck pendulum.

The first two are oscillatory type pendulums, the third - rotational.

Rotation and moment of inertia

When a body with a certain mass begins to rotate around an axis, its motion is usually described by the following equation:

Here M is the total, or resultant, moment of all external forces that act on the system, I is its moment of inertia and α is the angular acceleration.

M by definition is a quantity equal to the product of the acting force and the arm, which is equal to the distance from the point of the applied force to the axis of rotation.

The moment of inertia is a quantity that characterizes the inertial properties of the system, that is, how quickly it can be spun by applying a certain moment M. I also characterizes the kinetic energy stored by the rotating system. The moment of inertia I for a material point (an imaginary object whose mass is concentrated in an infinitesimal volume of space) performing circular motion at a distance from the r axis can be calculated using the following formula:

In the general case, when determining I for a body of arbitrary shape, one should use the following expressions:

1) I = ∑m i *r i 2 .
2) I = ∫dm *r i 2 = ρ*∫dV *r i 2 .

The first equality applies to a discrete arrangement of masses in the system, the second - to a continuous one.

From these expressions it is clear that I is a function of the distance to the axis of rotation and the distribution of mass in the system relative to this axis and does not depend on either the applied moments of force M or the rotation speed ω.

Mathematical (simple) pendulum

Since this type of oscillatory system is the simplest, we will consider it in more detail. A mathematical pendulum is a material point suspended on a weightless and inextensible thread. If this point is slightly deviated from the equilibrium position and then released, it will begin to oscillate. It is also assumed that there are no frictional forces at the point of attachment of the thread, and air resistance is neglected.

As is clear from the description above, a mathematical pendulum represents an ideal case that is not realized in practice. Nevertheless, its study allows us to obtain some important conclusions for the type of movement under consideration.

The figure below shows this pendulum, and also indicates the forces acting in the system when it oscillates.

Applying the equation of motion to it, we obtain the following equality:

M = -m*g*sin(θ)*L; I = m*L 2 ; α = d 2 θ/dt 2 =>
=> -m*g*sin(θ)*L = m*L 2 *d 2 θ/dt 2, from where:
L *d 2 θ/dt 2 + g*sin(θ) = 0.

Let us explain some of the thread tension T (see figure) is equal to zero, since it acts directly on the axis; the moment due to gravity is taken with a minus sign, since it is directed clockwise; L - thread length; angular acceleration α, by definition, is the second derivative of the rotation angle with respect to time or the first derivative with respect to time of the angular velocity ω; the formula for the moment of inertia of a pendulum of this type coincides with that for a material point with mass m located at a distance L from the axis of rotation.

The above expression can be simplified if we accept the approximation: sin(θ)≈θ. This is true when the vibration angles are small (up to θ=10 o the error does not exceed 0.5%). In this case we get:

L*d 2 θ/dt 2 + g*θ = 0.

We have obtained a classical differential equation (diff. equation) of the second order. Its solution is the sine function:

θ = A*sin(ω*t+θ 0).

Here A and θ 0 are the amplitude of oscillations and the initial angle of deviation from equilibrium, respectively. If this solution is substituted into the differential. ur. higher, then we can obtain the angular velocity and period of oscillation:

ω = √(g/L) and T = 2*pi/ω = 2*pi*√(L/g).

We obtained an amazing result: the period of oscillation of a mathematical pendulum does not depend on the initial conditions (A and θ 0), as well as on the mass m.

Galileo first began to study the behavior of a mathematical pendulum. Subsequently, Huygens showed the possibility of using the resulting formula to determine the acceleration of the Earth's free fall.

General type physical pendulum

This device is a solid body of arbitrary shape (its mass may be unevenly distributed over its volume), which oscillates about a horizontal axis that does not pass through the center of mass of the body.

When solving the equation of motion of this device, an ideal object is considered, the mass of which is concentrated at its center of gravity. This assumption leads to the following formula for the period of its oscillation:

T = 2*pi*√(I o /(m*g*h)).

Here h is the distance from the center of gravity to the axis of rotation O, I o is the moment of inertia of the physical pendulum. Note that if to calculate the moment of gravity you can use the additivity property of this quantity and reduce the sum of all moments to one applied to the center of gravity, then to calculate the moment of inertia I o you cannot do this; it should be calculated using the general formulas that were given previously.

Oscillating rod and its moment of inertia

Let's imagine that there is a solid rod of mass m and length L, which is suspended vertically from one of the ends. This design is capable of oscillating under the influence of gravity.

If we apply integration about the axis to such a rod, we can obtain that the moment of inertia of the pendulum of the physical structure indicated will be equal to:

Then its oscillation period will be equal to:

T = 2*pi*√(2*L /(3*g)).

The figure below shows this type of pendulum.

From the figure you can see that if you hang a weight on a thread, then 4 rods with weights begin to rotate with some angular acceleration.

The Oberbeck pendulum is used for laboratory work in physics to test the equation of rotational motion.

Determination of the moment of inertia of the Oberbeck pendulum

To solve this problem, it is necessary to make an important approximation: the weight of the rods and disks from which the overloads are suspended on a thread is negligible compared to the weight of one load m. Considering that the size of the loads is much smaller than their distance to the axis of rotation, we can use the formula for the moment of inertia of a material point. Since there are 4 weights and they all have the same mass, but are located at different distances from the axis, we obtain the following formula for the moment of inertia of the Oberbeck pendulum:

I = I 1 +I 2 +I 3 +I 4 = m*(R 1 2 +R 2 2 +R 3 2 +R 4 2).

Since this pendulum allows you to adjust the position of each weight on the rod, its moment of inertia can change.

Wind the suspension thread around the pendulum axis and secure it.

Check whether the lower edge of the ring corresponds to the zero of the scale on the column. If not, unscrew the top bracket and adjust its height. Screw the top bracket.

Press the “START” button of the millisecond watch (cell phone).

At the moment the pendulum passes the bottom point, stop the millisecond watch.

Wind the suspension thread around the pendulum axis, making sure that it is wound evenly, one turn next to the other.

Fix the pendulum, making sure that the thread in this position is not too twisted.

Record the measured value of the time the pendulum falls.

Define timing n= 10 times.

Determine the value of the average time of fall of the pendulum using the formula:

Where n– number of measurements taken, t i– time value obtained in i- that freeze, t– the average value of the time the pendulum falls.

Using the scale on the vertical column of the device, determine the distance covered by the pendulum during the fall.

Using formula (11) and known diameter values d o And d n, determine the diameter of the axis along with the thread wound around it.

Using formula (10), calculate the mass of the pendulum together with the ring imposed in this experiment. The mass values of individual elements are plotted on them.

Using formula (9), determine the moment of inertia of the pendulum.

Compare with theoretical value of moment of inertia

I theory = I o + I m,

Where I o– moment of inertia of the axis, I m- moment of inertia of the flywheel, which are calculated using the following formulas:

I o = m o r o 2 / 2; I k = m m r m 2 / 2 .

Practical data:

Pendulum length.

Table 1.

l, m	t1	t2	t3	t4	t5

Substituting everything and calculating we get:

I 1 =(0.00090±0.00001) kg*m2.

Conclusion: During the work, the moments of inertia of the pendulum were determined for different lengths of the wound thread and the errors were determined. A comparison of the calculated results and the experimental value reveals a significant difference in the data.

Conclusion: We have determined the experimental and theoretical moments of inertia of the pendulum, which amounted to

and compared them

1.1. The motion of Maxwell's pendulum is an example of the plane motion of a rigid body, in which the trajectories of all its points lie in parallel planes. This motion can be reduced to the translational motion of the pendulum and rotational motion around an axis passing through its center of mass perpendicular to these planes.

This type of motion is widespread in technology: the rolling of a cylinder on a plane, the wheels of a car, the roller of a road car, the movement of a rotating helicopter propeller, etc.

1.2. The purpose of this laboratory work is to experimentally familiarize ourselves with the plane motion of a rigid body using the example of a Maxwell pendulum and to determine the moment of inertia of the pendulum.

2. BASIC CONCEPTS

2.1. The Maxwell pendulum is a small flywheel. It can be lowered under the influence of gravity and the tension force of threads pre-wound on the axis of the pendulum (Fig. 1). The threads unwind completely during the downward movement. The untwisted flywheel continues to rotate in the same direction and winds the threads around the axis, as a result of which it rises up, while slowing down its movement. Having reached the top point, it begins to go down again.

The flywheel makes a periodically repeating motion, which is why it is called a pendulum. So, the movement of a Maxwell pendulum can be divided into two stages: lowering and rising.

2.2. According to the basic laws of the dynamics of translational and rotational motion (for the corresponding axes), neglecting the forces of friction against the air and the deviation of the threads from the vertical, we write

Where m- mass of the pendulum, I- moment of inertia of the pendulum relative to the axis, - pendulum axis radius, N- tension force of each thread, g- free fall acceleration, a- linear acceleration of the center of mass of the pendulum, - angular acceleration. Due to the inextensibility of the threads

These equations apply to both the first and second stages of the pendulum's motion. The initial conditions at different stages are different: when the pendulum is lowered, the initial speed of its center of mass is zero, and when it rises, it is different from zero.

2.3. From equations (1), (2), (3) it follows

(5)

From the dependence of the path on time for uniformly accelerated motion with zero initial speed, one can find the linear acceleration of the pendulum

Where t- time of movement of the pendulum from the top to the bottom point, h- the distance traveled during this time. At we have ; (7)

Note that the directions of linear acceleration and tension forces do not depend on whether the pendulum is moving up or down. During one complete oscillation, the linear velocity changes its direction at the bottom point to the opposite, but the linear acceleration and forces do not change. Angular velocity, on the contrary, does not change its direction, but the moment of force and angular acceleration at the bottom point are reversed.

2.4.When rising upward, the pendulum moves equally slow. Height h2, to which he rises will be less than the one from which he descends h1. The difference in these heights determines the decrease in mechanical energy spent on overcoming the forces of deformation of the threads upon impact and the forces of resistance to movement.

Proportion of lost mechanical energy

(9)

INSTALLATION DESCRIPTION

3.1. The installation diagram is shown in Fig. 2. A column 2 is fixed to the base 1; the upper bracket 3 is supported on it, on which there is an electromagnet 4, a photoelectric sensor 5 and a knob 6 for leveling the pendulum suspension. A second photoelectric sensor 7 is attached to the lower bracket. The Maxwell pendulum flywheel consists of a disk 8 mounted on an axis 9 and a massive ring 10 attached to it. It is suspended on two parallel threads wound on the axis. The pendulum is held in the upper position by an electromagnet. The heights of lowering and raising the pendulum are determined using a millimeter ruler 11 located on the column of the device. Millisecond watch MS 12 is designed for measuring time t movements of Maxwell's pendulum. The start and end of the time counting are carried out automatically using the photo sensors mentioned above.

The moment of inertia of a Maxwell pendulum is determined indirectly.

From equations (6) and (8) it follows that the moment of inertia can be calculated using the formula

Here m– total mass of the pendulum,

m = m O+m d+mK , (11)

Where m O - axle mass, m d - mass of the disk.

4. ORDER OF MEASUREMENTS

4.1. Technical data.

4.1.1. Enter the installation data into the table. 1.

Table 1

4.1.2. Enter into the table. 2 values of masses and diameters of pendulum elements. These data are indicated on the installation.

Table 2

4.3. Determination of the moment of inertia of a Maxwell pendulum.

4.2.2. Wind the suspension threads onto the pendulum axis symmetrically, turn to turn, and fix the pendulum. You should work very carefully.

4.2.3. Release the pendulum and start counting time. Stop the countdown at the bottom point.

4.2.5. Enter the measured value of the time of movement of the pendulum in Table 3. Repeating the operations in paragraphs 4.2.2 and 4.2.3, measure the time 10 more times and enter the data in the table. 3.

Table 3

4.3. Determination of loss of mechanical energy

4.3.1. Use a ruler to determine the height h 1, from which the pendulum descends; enter into the table 3.

4.3.2. Repeat the operations described in paragraphs 4.2.2 and 4.2.3, let the pendulum perform five full oscillations, measure the height difference d h. Perform this measurement once and enter its result in the table. 3.

5. PROCESSING OF MEASUREMENT RESULTS

5.1. Determination of the moment of inertia of a Maxwell pendulum.

Calculate the average value of the time of movement of the pendulum and enter it in the table. 3.

Calculate the mean square error in measuring the time of movement of the pendulum

(12)

5.1.3. Calculate absolute random error

D t sl = 2,1D.S.. (13)

5.1.4. Calculate the total absolute error

D t = D t сл + D t inc.(14)

5.1.5. Calculate relative error

Place all calculated values in the table. 3.

5.1.6. Using formula (10), calculate the moment of inertia of the pendulum, substituting its average value.

5.1.7. Calculate the relative error of the moment of inertia of the pendulum

, (16)

Where D m , D r O, D h1- instrument errors of the corresponding quantities, Dt – total absolute error of movement time; m- the total mass of the pendulum, calculated using formula (11).

5.1.8. Based on the received value eJ calculate the absolute error value DJ in determining the moment of inertia

DJ = e J J= . (17)

Round DJ to one significant figure, and the values `J to the level of absolute error.

5.1.9. Write the final result in the form

J =`J± D J =(±) kg × m 2 . (18)

5.2. Determination of the loss of mechanical energy during the movement of a Maxwell pendulum.

5.2.1. Formula (9) expresses the fraction of mechanical energy lost during five oscillations of the Maxwell pendulum; for one oscillation the share will be five times less:

6. QUESTIONS submitted for JOB DEFENSE

1. The basic law of the dynamics of translational motion.

3. How do the momentum and axial angular momentum of a Maxwell pendulum change at the lowest point of its motion? Explain your reasons.

4. Law of conservation of total energy for Maxwell's pendulum.

5. Find the linear and angular velocities of the pendulum at the lowest point.

6. Moment of inertia of a rigid body (definition). What does its size depend on?

7. Find the ratio of the kinetic energy of translational motion to the kinetic energy of rotational motion for a given Maxwell pendulum.

8. How do linear and angular accelerations change during the period of motion of the Maxwell pendulum?

9. Momentum and axial angular momentum of a rigid body.

10. Estimate the tension of the threads when the pendulum passes the lowest point (take the duration of the “blow” equal to Dt"0.05c).

11. How will the time of movement of the pendulum change if the radius of its axis is doubled?

12. Kinetic energy of translational and rotational motion of a rigid body.

13. Calculation of the moment of inertia of a disk with a radius R, mass m

14. What forces and moments of forces act on the Maxwell pendulum during its movement? How do they change over the period?

15. Calculation of the moment of inertia of a ring with a radius R, mass m relative to an axis passing through the center perpendicular to its plane.

16. Obtain formula (10) based on the law of conservation of mechanical energy. (Please note that for the Maxwell pendulum E to vr >>E to post).

17. In which part of the pendulum’s motion, upper or lower, is the loss of mechanical energy greater? Explain the reasons.

DETERMINATION OF MOMENT OF INERTIA

PHYSICAL PENDULUM

Purpose of the work: familiarization with the physical pendulum and determination of its moment of inertia relative to the axis of rotation. Study of the dependence of the magnitude of the moment of inertia of a pendulum on the spatial distribution of mass.

Devices and accessories: a physical pendulum with a bracket for its suspension, a metal prism for determining the position of the center of gravity of the pendulum, a stopwatch.

Theoretical introduction.

A physical pendulum (Fig. 1) is any rigid body that, under the influence of gravity, oscillates around a fixed horizontal axis (O) that does not pass through its center of gravity (C). The pendulum's suspension point is the center of rotation.

Fig.1. Physical pendulum

When the pendulum deviates from the equilibrium position by an angle , a torque created by gravity occurs:

Where l– the distance between the suspension point and the center of gravity of the pendulum (the minus sign is due to the fact that the moment of force M has such a direction that it tends to return the pendulum to the equilibrium position, i.e. decrease angle ).

For small deflection angles
, Then

(0)

On the other hand, the moment of the restoring force can be written as:

(0)

I– moment of inertia of the pendulum

i– angular acceleration.

From (1) and (2) we can obtain:

Designating
(0)

we get
(4)

Equation (4) is a 2nd order linear differential equation. Its solution is the expression
.

Taking into account equation (3), the period of small oscillations of a physical pendulum can be written as:

, (5)

Where
- reduced length of the physical pendulum

From formula (5) we can express the moment of inertia of a physical pendulum relative to the axis of rotation

(6)

Finding by measurements m, l And T, you can use formula (6) to calculate the moment of inertia of a physical pendulum relative to a given axis of rotation.

In this work, a physical pendulum is used (Fig. 2), which is a steel rod on which two massive steel lentils (A 1 and A 2) and support prisms for suspension (P 1 and P 2) are fixed. The moment of inertia of such a pendulum will be the sum of the moments of inertia of the rod, lentils and prisms:

Where I 0 - moment of inertia of the rod relative to the axis passing through the center of gravity.

(7)

m st– mass of the rod,

l st– length of the rod,

d– distance from the center of gravity of the rod to the suspension point.

The moments of inertia of lentils and prisms can be approximately calculated as for point masses. Then the moment of inertia of the pendulum will be written as:

Where
- masses of lentils A 1 and A 2,

- distances from the axis of rotation (suspension point) to lentils A 1 and A 2, respectively,

- masses of prisms P 1 and P 1,

- distances from the axis of rotation to prisms P 1 and P 2, respectively.

Because according to the conditions of the work, only one lentil A 1 moves, then only the moment of inertia will change And

(9)

Description of installation.

The physical pendulum used in this work (Fig. 2) is a steel rod (C), on which two massive steel lentils (A 1 and A 2) and support prisms for suspension (P 1 and P 2) are fixed. The pendulum is suspended on a bracket.

By moving one of the lentils, you can change the moment of inertia of the pendulum relative to the suspension point (axis of rotation).

The center of gravity of the pendulum is determined by balancing the pendulum on the horizontal edge of a special prism (Fig. 3). On the pendulum rod, ring grooves are applied every 10 mm, which serve to accurately determine the distance from the center of gravity to the axis of rotation without the help of a ruler. By slightly moving the lentil A 1 along the rod, you can achieve the distance l from the point of suspension to the center of gravity was equal to a whole number of centimeters, measured on the scale on the rod.

The order of work.

Determine the position of the center of gravity of the pendulum.

A ) Remove the pendulum from the bracket and install it in a horizontal position on a special prism P 3 (Fig. 3) so that it is in balance. The exact equilibrium position is achieved by slightly moving the lentil A 1 .

Fig.3. Balancing the pendulum

b) Measure on the scale on the pendulum l - the distance from the suspension point (prism edge P 1) to the center of gravity of the pendulum (upper edge of prism P 3).

c) Measure the distance using the pendulum scale - from the suspension point (prism edge P 1) to the upper lentil A 1.

2. Determine the period of oscillation of a physical pendulum.

a) Install the pendulum with prism P 1 on the bracket (Fig. 2)

b) Determine the time of complete 50 - 100 oscillations of the pendulum. Record time t and number n pendulum oscillations.

c) Determine the period of oscillation of a physical pendulum using the formula:

(10)

3. Remove the pendulum from the bracket. Move lentil A 1 a few centimeters to a new position and repeat the experiment. Measurements must be made for at least three different positions of the lentil A 1 relative to the suspension point.

4. Using formula (6), calculate the moment of inertia of the physical pendulum I op .

5. Calculate the relative error of the moment of inertia for one of the considered cases using the formula:

. (11)

Values  T And  l determined by the accuracy class of the instruments.

6. Find the absolute error
for each case, taking the relative error the same for all cases.

Write the final result in the table in the form

7. Using formula (8), calculate the moment of inertia of the pendulum I theory for every occasion.

8. Compare the results obtained I op And I theory, calculating the ratio:

(12)

Draw a conclusion about how large the discrepancy between the obtained values is and what are the reasons for the discrepancies.

Results of measurements and calculations

№ p/p					,	, kg m 2	I theory, kg m 2

Test questions.

What is a physical pendulum?

What is the reduced length of a physical pendulum?

What vibration is called harmonic?

What is an oscillation period?

Derive a formula to calculate the period of oscillation of a physical pendulum.

What is moment of inertia? What is the additivity of the moment of inertia?

Obtain a formula for calculating the moment of inertia of a physical pendulum.

Literature

1. Savelyev I.V. Course of general physics: Textbook. manual for colleges: in 3 volumes. T.1: Mechanics. Molecular physics. - 3rd ed., rev. - M.: Nauka, 1986. – 432 p.

2. Detlaf A. A., Yavorsky B. M. Physics course: Textbook. allowance for colleges. - M.: Higher School, 1989. - 607 p. - subject decree: p. 588-603.

3. Laboratory workshop in physics: Proc. manual for college students / B. F. Alekseev, K. A. Barsukov, I. A. Voitsekhovskaya and others; Ed. K. A. Barsukova and Yu. I. Ukhanova. – M.: Higher. school, 1988. – 351 p.: ill.

METHODOLOGICAL INSTRUCTIONS FOR LABORATORY WORK No. 1.2

DETERMINATION OF THE MOMENT OF INERTIA OF A PHYSICAL

PENDULUM

PURPOSE OF THE WORK: to determine the moment of inertia of a physical pendulum and to study the dependence of the moment of inertia on the position of the center of mass of the pendulum relative to the axis of rotation.

DEVICES AND ACCESSORIES: physical pendulum on a bracket, stopwatch, prism on a stand, scale ruler.

ELEMENTS OF THE THEORY

Periodic displacements of a body relative to some stable position (equilibrium position) are calledoscillatory movement or simple vibrations. Oscillatory movements in general represent complex physical processes. The study of vibrations serves as the basis for a number of applied disciplines (acoustics, machine theory, seismology, etc.).

The simplest type of oscillation is harmonic oscillatory motion. Harmonic vibrations of a body occur when a force is applied to it that is proportional to the displacement, i.e.. This force is called quasi-elastic or restoring. The nature of the restoring force can be different (elastic force, gravity, etc.) With harmonic motion, the dependence of the path (displacement) from time expressed by the sine or cosine function:

Where maximum displacement of the body from the equilibrium position (amplitude),

circular or cyclic frequency,

Time of one complete oscillation (period),

initial phase of oscillation.

The acceleration of a body performing harmonic oscillations is proportional to the displacement and is always directed towards equilibrium, i.e. for each moment of time offset and acceleration have opposite signs:

. (1)

Harmonic oscillations are performed by pendulums under the influence of gravity if the angles of deviation from the vertical position (equilibrium position) are small.

Pendulums can be simple or complex. A small body (material point) suspended on a long thread, the tension and weight of which can be neglected, is called simple ormathematical pendulum. A solid body of arbitrary shape, fixed on a horizontal axis that does not pass through the center of gravity, is a complex orphysical pendulum.

Any solid body can be considered as a collection of invariably connected material points with masses, , . . ., , therefore the moment of inertia of a physical pendulum can be defined as the sum of the moments of inertia of all its material points:

, (2)

where r the distance from each of them to the axis of rotation.

In practice, it is not possible to use formula (2), therefore, to determine the moment of inertia of a physical pendulum, we will describe its oscillations using the law of the dynamics of rotational motion.

There are two forces acting on a physical pendulum: the force of gravity applied to the center of gravity of the pendulum (point), and the support reaction force applied at the point where the pendulum is attached, where the axis of rotation passes.

When a physical pendulum deviates from its equilibrium position by an angle(Fig. 1) gravity will create a torque, under the influence of which vibrations will begin.

Rice. 1

The moment of gravity determines the angular acceleration.

If we denote the distance from the axis of rotationto the center of gravity through , then the moment of gravity would be expressed like this:

or at small angles

, (3)

Where gravity arm, pendulum mass, acceleration of free fall of a body. “-” is explained by the restoring nature of the moment of force. It is directed opposite to the pendulum's deflection angle.

When a pendulum oscillates, its center of gravity moves along an arc of a circle, so its movement can be described using the law of the dynamics of rotational motion. It will be written in the form:

, (4)

Where moment of inertia of the body about the axis of rotation.

Substituting into equation (4) the value(3) and solving it with respect to angular acceleration, we obtain

, (5)

Equation (5) differs from equation (1) only in that it includes angular quantities instead of linear ones.

From a comparison of equations (1) and (5) it follows that or , from which we obtain the formula for the period of oscillation of a physical pendulum:

. (6)

From the formula for the period of oscillation of a physical pendulum (5) we find its moment of inertia:

, (7)

Where period of oscillation of the pendulum.

This expression is a calculation formula for determining the moment of inertia of a physical pendulum.

EXPERIMENTAL METHOD AND DESCRIPTION OF THE INSTALLATION

The physical pendulum in this work consists of a steel rod O D , on which a massive cylindrical body B is attached with screws (Fig. 2). When the support screws are released, body B can be moved along the rod and, therefore, change the position of the center of gravity of the pendulum.

To suspend the pendulum, use a special bracket on which the pendulum is suspended at the point.

Rice. 2

Rice. 3

To find the center of gravity of the pendulum (point) is a special prism mounted on a stable stand (the edge of a chair). The pendulum is placed horizontally on the edge of this prism and, observing the balancing, a position is found in which the moments of gravity acting on the right and left parts of the pendulum will be equal (Fig. 3). In this position, the center of gravity of the pendulum will be located in the rod opposite the fulcrum. Distancedetermined using a scale bar.

PROCEDURE FOR PERFORMANCE OF THE WORK

Determine the total mass of the pendulum (rod and load) in kilograms.
Having strengthened the load B at the end of the rod, determine the position of a point on any supportand measure the distance r scale ruler.
Having hung the pendulum on the bracket, deflect it from the equilibrium position by a small angle (the end of the rod is retracted to a distance of 6-8 cm) and release it. After skipping 3-4 full swings, start the stopwatch at the moment when the pendulum reaches its maximum deviation. Determine the time3050 full swings of the pendulum ().
Repeat the operation described in paragraph 3 3 more times and, using the data obtained, determine the average value of the period of oscillation of the pendulumat a given position of the load.
Move the load along the rodby 6-7 cm and repeat the described determination operations And at a new position of the load B.
The work ends if such movements of the load with accompanying measurements are done 3-5 times.
The obtained experimental data are substituted into formula (7) and the moments of inertia of the pendulum are calculated in the SI system of units at different distances of the center of gravity from the axis of rotation.
The results of measurements and calculations are recorded in the table:

Kg	kg m 2	Kg m 2	Kg m 2	Kg m 2



Kg	kg m 2	Kg m 2	Kg m 2	Kg m 2



Kg	kg m 2	Kg m 2	Kg m 2	Kg m 2

The results of moments of inertia are recorded in standard form (in the form of intervals).
Based on the results of the table, a conclusion is drawn about the dependence of the moment of inertia of a physical pendulum on the position of its center of gravity.

TEST QUESTIONS

What vibrations are called free?
What vibrations are called harmonic?
Write down the equation of free harmonic oscillations.
What is the frequency of oscillations, their period and amplitude?
What characteristics of harmonic oscillations do not change over time?
What characteristics of oscillations are harmonic functions of time?
Define the moment of inertia of a material point and the moment of inertia of a body.
Define a physical pendulum. How does the moment of inertia of a physical pendulum depend on the position of the cylinder on the rod?
Give me 2! determining the moment of force (through the distance from the center of gravity to the axis of rotation and through the arm of the force). How to determine the direction of a moment of force?
Write down the basic law of dynamics for rotational motion and get formula for the period of oscillation of a physical pendulum with accompanying explanations(use additional literature).

Educational institution

Department of Mathematics and Physics

PENDULUM

METHODOLOGICAL INSTRUCTIONS FOR LABORATORY WORK No. 1.2

by discipline

"PHYSICS"

Educational institution

"HIGHER STATE COLLEGE OF COMMUNICATIONS"

Department of Mathematics and Physics

DETERMINATION OF THE MOMENT OF INERTIA OF A PHYSICAL

PENDULUM

METHODOLOGICAL INSTRUCTIONS FOR LABORATORY WORK No. 1.2

by discipline

"PHYSICS"

for students of all specialties

DETERMINATION OF THE MOMENT OF INERTIA OF A PHYSICAL

PENDULUM

DEVICES AND ACCESSORIES: physical pendulum on a bracket, stopwatch, prism on a stand, scale ruler.

ELEMENTS OF THE THEORY

Periodic displacements of a body relative to some stable position (equilibrium position) are called oscillatory movement or simple vibrations. Oscillatory movements in general represent complex physical processes. The study of vibrations serves as the basis for a number of applied disciplines (acoustics, machine theory, seismology, etc.).

The simplest type of oscillation is harmonic oscillatory motion. Harmonic vibrations of a body occur when a force is applied to it that is proportional to the displacement, i.e. . This force is called restoring. The nature of the restoring force can be different (elastic force, gravity, etc.) With harmonic motion, the dependence of the path (displacement ) from time expressed by the sine or cosine function:

Where - maximum displacement of the body from the equilibrium position (amplitude),

- circular or cyclic frequency,

- time of one complete oscillation (period),

- initial phase of oscillation .

. (1)

Harmonic oscillations are performed by pendulums under the influence of gravity if the angles of deviation from the vertical position (equilibrium position) are small. Pendulums can be simple or complex. A small body (material point) suspended on a long thread, the tension and weight of which can be neglected, is called simple or mathematical pendulum. A solid body of arbitrary shape, fixed on a horizontal axis that does not pass through the center of gravity, is a complex or physical pendulum.

Any solid body can be considered as a collection of invariably connected material points with masses
,
, . . .,
.

When a physical pendulum deviates from its equilibrium position by an angle (Fig. 1) each of its elements will be affected by the moment of gravity relative to the axis of rotation . The sum of the moments of all these forces is equal to the moment of the resultant forces of gravity
, applied to the center of gravity of the pendulum (point ).

Under the influence of the moment of gravity, the pendulum begins to oscillate with angular acceleration
.

If we denote the distance from the axis of rotation to the center of gravity through , then the moment of gravity
would be expressed like this:

or at small angles

, (2)

Where - shoulder strength
,

- mass of the pendulum,

- acceleration of free fall of a body in a given place.

When a pendulum oscillates, its center of gravity moves along an arc of a circle, therefore the equation of Newton's second law for rotational motion is also applicable for a pendulum. It will be written in the form:

, (3)

Where ‑ moment of inertia of the body about the axis of rotation .

Moment of inertia material point is called the product of mass (
)per distance squared ( ) from the axis of rotation to it (
). The moment of inertia of a body is equal to the sum of the moments of inertia of its particles relative to the same axis, that is

Substituting into equation (3) the value
and solving it with respect to angular acceleration, we get

, (4)

Equation (4) differs from equation (1) only in that it includes angular quantities instead of linear ones.

From a comparison of equations (1) and (4) it follows that
or
, from which we obtain the formula for the period of oscillation of a physical pendulum:

. (5)

From the formula for the period of oscillation of a physical pendulum (5) we find its moment of inertia:

, (6)

Where
- period of oscillation of the pendulum.

This expression is a calculation formula for determining the moment of inertia of a physical pendulum.

EXPERIMENTAL METHOD AND DESCRIPTION OF THE INSTALLATION

The physical pendulum in this work consists of a steel rod

ОD, on which a massive cylindrical body B is attached with screws (Fig. 2). When the support screws are released, body B can be moved along the rod and, therefore, the position of the center of gravity of the pendulum can be changed.

To suspend the pendulum, use a special bracket on which the pendulum is suspended at the point .

To find the center of gravity of the pendulum (point ) is a special prism mounted on a stable stand. The pendulum is placed horizontally on the edge of this prism and, observing the balancing, a position is found in which the moments of gravity acting on the right and left parts of the pendulum will be equal (Fig. 3). In this position, the center of gravity of the pendulum will be located in the rod opposite the fulcrum. Distance
determined using a scale bar.

PROCEDURE FOR PERFORMANCE OF THE WORK

etc. For , And r 3.

Addiction from are depicted graphically in the selected coordinate system, and the value is plotted on the horizontal axis (m), and on the vertical (kg m 2 ).