Q1: The motion of the simple pendulum can be approximated by the simple harmonic motion (SHM) for small oscillations amplitude. In the SHM, the period does not depend on the amplitude of the oscillation. So, one can also say that the period of the simple pendulum is, roughly, independent of the amplitude, as long as the amplitude is "small". For larger amplitudes, the approximation becomes crude and the period considerably deviates from the SHM period. One can obtain different expressions for the simple pendulum period that differ in their accuracy in estimating the "true" period of the system. Some of them are listed below (assuming the length of the pendulum `= 9.8 m): To = 2, 7₁-2x (1+1) = T₂ T 2T √1-4²/8', where A is the amplitude of the oscillations. Clearly. To above gives the approximated SHM period. The exact period of the simple pendulum can be expressed as 4 - S. √1-sin² (4) sin² u du (a) Evaluate T₁ T2 and T for a simple pendulum that starts from rest with an initial angular displacement of 80 = 1, 10° and 20°. Compare your results with each other and with the SHM period. What is your comment? (b) If you tolerate a deviation of no more than 1% between the estimated period and the true one, what is the maximum initial angular displacement(s) that the pendulum can have so that its period can be adequately approximated by To, T₁ or T₂? (c) (Optional): Plot To. T₁, T2 and T as a function of A for amplitudes between 0 and 1/2 and discuss your results.
Q1: The motion of the simple pendulum can be approximated by the simple harmonic motion (SHM) for small oscillations amplitude. In the SHM, the period does not depend on the amplitude of the oscillation. So, one can also say that the period of the simple pendulum is, roughly, independent of the amplitude, as long as the amplitude is "small". For larger amplitudes, the approximation becomes crude and the period considerably deviates from the SHM period. One can obtain different expressions for the simple pendulum period that differ in their accuracy in estimating the "true" period of the system. Some of them are listed below (assuming the length of the pendulum `= 9.8 m): To = 2, 7₁-2x (1+1) = T₂ T 2T √1-4²/8', where A is the amplitude of the oscillations. Clearly. To above gives the approximated SHM period. The exact period of the simple pendulum can be expressed as 4 - S. √1-sin² (4) sin² u du (a) Evaluate T₁ T2 and T for a simple pendulum that starts from rest with an initial angular displacement of 80 = 1, 10° and 20°. Compare your results with each other and with the SHM period. What is your comment? (b) If you tolerate a deviation of no more than 1% between the estimated period and the true one, what is the maximum initial angular displacement(s) that the pendulum can have so that its period can be adequately approximated by To, T₁ or T₂? (c) (Optional): Plot To. T₁, T2 and T as a function of A for amplitudes between 0 and 1/2 and discuss your results.
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Transcribed Image Text:Q1: The motion of the simple pendulum can be approximated by the simple harmonic motion (SHM)
for small oscillations amplitude. In the SHM, the period does not depend on the amplitude of the
oscillation. So, one can also say that the period of the simple pendulum is, roughly, independent of the
amplitude, as long as the amplitude is "small". For larger amplitudes, the approximation becomes crude
and the period considerably deviates from the SHM period. One can obtain different expressions for the
simple pendulum period that differ in their accuracy in estimating the "true" period of the system. Some
of them are listed below (assuming the length of the pendulum `= 9.8 m):
To = 27,
A?
1+
16
T1 = 27
27
T2 =
VI- A²/8',
where A is the amplitude of the oscillations. Clearly, To above gives the approximated SHM period.
The exact period of the simple pendulum can be expressed as
*/2
4
T =
du
V1
– sin°(4) sin² u
(a) Evaluate T1. T2 and T for a simple pendulum that starts from rest with an initial angular
displacement of 0o = 1', 10' and 20'. Compare your results with each other and with the SHM
period. What is your comment?
(b) If you tolerate a deviation of no more than 1% between the estimated period and the true one,
what is the maximum initial angular displacement(s) that the pendulum can have so that its
period can be adequately approximated by To. T1 or T2?
(c) (Optional): Plot To. T1, T2 and T as a function of A for amplitudes between 0 and 1/2 and discuss
your results.
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