DETERMINATION OF THE RESULTED LENGTH OF PHYSICAL PENDULUM

Purpose of work: study of laws of oscillation physical and mathematical pendulums.

Task: experimentally and theoretically to find resulted length of physical pendulum.

Devices and equipments: physical pendulum, mathematical pendulum, straightedge, stop-watch. The experimental setting (fig.10.1) consists of physical 1 and mathematical 2 pendulums.

Figure 10.1

Length of mathematical pendulum can be fluently changed, reeling it in 5 and to fix spirally 6.

Theoretical part

A physical pendulum is a body which can be revolved relatively of arbitrary horizontal axis, that not go through the center of mass. Under the action of moment of force mg, the arm of which is L·sinα, a body is revolved round the point of hang up O (fig.10.2). L is distance from a point O rotation (points of hang up) to the point of C - center of mass of body. Write down the fundamental equation of the rotational motion dynamics

, (10.1) I is a moment of inertia of body, is angular acceleration. A sign does minus take into account, that the moment of force of mg is diminished by a corner α.

Figure 10.2

Thus, get differential equation of undamped oscillation of the physical pendulum

. (10.2)

At small corners α (less 5^о) is it possible, that sin α = α. Get

(10.3) Comparing this equation to general equation of undamped harmonic oscillations

, (10.4)

get cyclic frequency and period of oscillation of the physical pendulum

(10.5)

Thus, the period of oscillation of the physical pendulum depends on position of point of hang up O and forms of body, that to its moment of inertia in relation to this point.

For a mathematical pendulum, which is a material point, suspended on a weightless unstretching thread long L, moment of inertia is , . Consequently the period of oscillation of the mathematical pendulum depends only on length of thread

. (10.6)

The resulted length of L_res of physical pendulum is such length of mathematical pendulum the period oscillation of which equals the period of oscillation of the physical pendulum. From (10.5) and (10.6) we have

. (10.7)

A moment of inertia of peg (fig.10.1) is taking into account a theorem Steiner

. (10.8)

Thus, from (10.7) and (10.8) get the theoretical value of the resulted length

. (10.9)

Find the theoretical value of period of oscillation of the physical pendulum from (10.5) and (10.8)

. (10.10)

Practical part

1. Take off a physical pendulum from a bracket.
2. By a line to measure general length of b peg.
3. Set a supporting prism 3 in the distance and a = 20 + N of see from his middle and to fix its spirally 4. N is a number of educational brigade.
4. Hang up a physical pendulum.
5. Decline mathematical and physical pendulums on a corner approximately 5^о and to release.
6. By sight to watch after synchronousness of oscillations of both pendulums. In case, if the period of mathematical pendulum more than (less) physical, to decrease (to increase) length of thread of mathematical pendulum.
7. To repeat points 5, 6 to coinciding of periods of oscillations of pendulums, that synchronous oscillation during not less than 20 oscillations.
8. To measure time t of 20 oscillations of pendulums and to find a period .
9. By a line to measure length of mathematical pendulum from the point of hang up to the center of peg. It will be experimental value of the resulted length of physical pendulum of L_exp .
10. Expect the theoretical values of the resulted length after a formula (10.8) and period after a formula (10.10).
11. Compare the experimental and theoretical values of resulted length and to the period, writing down them in a table 10.1.

Table 10.1

12. Expect the error of L_theory.

Control questions

8. What is physical pendulum?

9. What is mathematical pendulum?

10. Get differential equation of free harmonic oscillation of the physical pendulum.

11. Get the period of oscillation of the physical pendulum.

12. Get the period of oscillation of the mathematical pendulum.

13. Give determination of the resulted length of physical pendulum.

14. Get expression for the resulted length of physical pendulum.

Literature

1 Чолпан П.П. Фізика.- К.: Вища школа, 2003.- С.77-80.

2. Лапотинський І.Е., Зачек І.Р. Фізика для інженерів.- Львів: Афіша, 2003.-С.

3. Савельев И.В. Курс общей физики. - т.1, М.: Наука,1982.- С.196-199.

4. Трофимова Т.И. Курс физики.- М: Высшая школа, 1990.- С.222-223.

Translator: S.P. Lushchin, the reader, candidate of physical and mathematical sciences.

Reviewer: S.V. Loskutov, professor, doctor of physical and mathematical sciences.

Approved by the chair of physics. Protocol № 6 from 30.03.2009 .

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