potential energy V(x). An alternate way of approaching this problem is by directly considering the restoring torque 7: T= -mgl sin 0. Angles and lengths are as depicted in the figure below. \ \ \\ Gravitational restoring torque t: Expand out 7(0) as a power series to third order in 0.
potential energy V(x). An alternate way of approaching this problem is by directly considering the restoring torque 7: T= -mgl sin 0. Angles and lengths are as depicted in the figure below. \ \ \\ Gravitational restoring torque t: Expand out 7(0) as a power series to third order in 0.
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Question
Transcribed Image Text:In Problem Set 2a you analyzed a pendulum by expanding out the gravitational
potential energy V (x). An alternate way of approaching this problem is by directly
considering the restoring torque T:
T= -mgl sin 0.
Angles and lengths are as depicted in the figure below.
Gravitational
restoring
torque t:
(a) Expand out T(0) as a power series to third order in 0.
(b) Using the information from your answer to (b), determine whether the period of this
pendulum at high amplitudes 69 is longer, shorter, or the same as the period at low
amplitudes. Hint: The dynamics of this problem are governed by Newton's second law for
angular motion T = Ia = ml²6, which is to say that the torque is directly proportional to
the pendulum's instantaneous angular acceleration.
Expert Solution
Step 1
(a)
We have been provided with the expression for restoring torque as-
According to the question,
we have to expand the in the series of third power of
Hence,
We will expand the sin in power series as shown below,
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