5. Kater\'s Pendulum PDF

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Determinig time period of Kater's pendulum...

Description

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Reversible pendulum (Kater’s pendulum)

Objective: To determine the value of ‘g’ at NISER using a reversible pendulum. Apparatus: A reversible pendulum, knife edges, support stands, photodetector/stop watch, measuring tape/scale. Theory: A physical pendulum is a rigid body swing in a vertical plane about any horizontal axis passing through the body. The resultant force acts through the centre of mass. The time period of oscillation of the physical pendulum is related to the moment of inertia I about the point of suspension.

ܶ ൌ ʹߨට

ூ

ெ௚௟

(1)

Where M is the mass of the rigid body and l is the separation between the point of suspension and the centre of gravity. Using the parallel axis theorem the moment of inertia I can be expressed in terms of the moment of inertia about the centre of gravity as

‫ ܫ‬ൌ ‫ ݇ܯ‬ଶ ൅ ‫ ݈ܯ‬ଶ

A

(2)

lA

Where k is the radius of gyration and this lead to an expression

ܶ ൌ ʹߨට

௞ మ ା௟ మ

CG lB

௚௟

(3)

B

The reversible pendulum (Kater’s Pendulum) has two pivot points on opposite sides of the centre of gravity from which the pendulum can be suspended (as shown in Fig. 1). lA and lB are the distance of the centre of gravity from pivot points A and B, respectively. Following Eq.(3), the time period about A and B can be written as

ܶ஺ ൌ ʹߨට

௞ మ ା௟ಲమ ௚௟ಲ

and

ܶ஻ ൌ ʹߨට

Fig. 1

మ ௞ మ ା௟ಳ ௚௟ಳ

If we adjust the length lA and lB such that TA = TB = T then by simple manipulation we can obtain

ܶ ൌ ʹߨට

ሺ௟ಲ ା௟ಳ ሻೝ ௚

1 

(4)

݃ ൌ Ͷߨ ଶ

ሺ௟ಲ ା௟ಳ ሻೝ ்మ

(5)

Here ሺ݈஺ ൅  ݈஻ ሻ௥ is the equivalent length of the pendulum, which satisfies the condition of reversibility or in other words the period of oscillation around bearing sleeve A and B become equal. Thus, Eq.(5) eliminates the need to locate the center of gravity or the radius of gyration of the rigid body, which is a major of error inherent to the simple pendulum and bar pendulum. The Setup The experimental set-up is illustrated in Fig.2. The knife edges must be fixed at the same height so as to make sure the mass of the pendulum will be distributed evenly over both bearing points. Make sure that the table and the stage supporting the knife edges do not move with the pendulum. The geometrical length of the rod used as reversible pendulum is 75 cm to which the bearing sleeves (referred as A and B) can be screwed at a desired position. The time period T of the pendulum is determined for small oscillating amplitudes with the fork light barrier. The light barrier is operated in the “period measurement” mode (switch shifted to the right side) and situated at the point of maximum amplitude of the pendulum. The time elapsed between two consecutive phases is measured, when the pendulum just leaves the infrared beam. This lapse of time is the period T at the point of maximum amplitude, as long as the pendulum keeps covering the beam while it reverses its trajectory.

Fig. 2

Procedure:

In this experiment in order to find g, first we need to determine the equivalent length ሺ݈஺ ൅  ݈஻ ሻ௥ . 1. Throughout the experiment sleeve A is fixed at a distance of 7 – 10 cm from one end of the pendulum. Position of sleeve B can be varied. 2. Adjust the bearing sleeve B so that the distance between the two sleeves, ሺ݈஺ ൅  ݈஻ ሻ is roughly 45cm. Make sleeve A as the pivot point and record the time period TA about A. 3. Now turn the pendulum upside down and make sleeve B as the pivot point. Measure the time period TB by varying the position of sleeve B (keeping position of sleeve A fixed) such that ሺ݈஺ ൅  ݈஻ ሻ is between 34 cm and 60 cm, in steps of 2 cm (just because we already know that this range will work). 4. Plot the time period TB as a function of ሺ݈஺ ൅  ݈஻ ሻ. Find the length ሺ݈஺ ൅  ݈஻ ሻ௥ from the graph corresponding to T = TB = TA, as recorded in step-2. See Fig. 3 for reference.

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5. Now to determine the value of T and ሺ݈஺ ൅  ݈஻ ሻ௥ more accurately, record both TA and TB about sleeves A and B as pivot points, respectively. Again, position of sleeve A remains fixed and position of sleeve B is varied in the range ሺ݈஺ ൅  ݈஻ ሻ௥ ± 3 cm with smaller intervals of 0.3 cm. 6. Plot TA and TB as a function of ሺ݈஺ ൅  ݈஻ ሻ on the same scale as shown in Fig. 4. Find T and ሺ݈஺ ൅  ݈஻ ሻ௥ at point of intersection of the two curves and calculate ‘g’ using Eqn.(5).

TB(sec)

TA(sec)

(lA+lB)r

lA+lB(cm)

Fig. 3 Observations: (I) (II)

ሺ݈஺ ൅  ݈஻ ሻ = 45 cm, TA = ………… sec

Table 1: Data for TB ~ ሺ݈஺ ൅  ݈஻ ሻ graph No. of Obs. ሺ࢒࡭ ൅ 1 2 .. .. ..

(III)

࢒࡮ ሻ (cm) TB (sec) 34 .. 36 .. .. .. .. .. 60 ..

Table 2: Data for TA, TB ~ ሺ݈஺ ൅  ݈஻ ሻ graph

No. of Obs. ሺ࢒࡭ ൅ ࢒࡮ ሻ (cm) TA (sec) TB (sec) .. .. .. .. 3

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.. .. .. .. Graphs/Calculations:

.. .. .. .. g = …………. cm/sec2

.. .. .. ..

.. .. .. ..

Computation of proportional error: ߜܶ ߜሺ݈஺ ൅  ݈஻ ሻ௥ ߜ݃ ൰ ൅  ൬ʹ ൰ ൌ  ඨ൬ ܶ ሺ݈஺ ൅  ݈஻ ሻ௥ ݃ ଶ

ଶ

Precautions: References: 1. Practical physics: R.K. Shukla, Anchal Srivatsava 2. Kenneth. E. Jesse, American Journal of Physics, Vol. 48, Issue 9, pp.785 (1980)

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