mass m is fixed to Pendulum A simple pendulum consisting of a point 1. the end of a massless rod (length /), whose other end is pivoted from the ceiling to let it swing freely in a vertical plane, specified by its angle o from the equilibrium position. (a) Prove that the pendulum's potential energy (measured from the equilibrium level) is as shown below. The pendulum's position can be U ()mgl (1 - cos ) (b) Write down the total energy E as a function of and o. Show that by differentiating your expression for E with respect to t you can get the equation of motion for ¢ and that the equation of motion is just the familiar T moment of inertia, and a is the angular acceleration ). (c) Assuming that the angle remains small throughout the motion, show that the motion is periodic with period Ia (where T is the torque, I is the - 2т V1/9. Tо m 1

University Physics Volume 1
18th Edition
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Author:William Moebs, Samuel J. Ling, Jeff Sanny
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Chapter7: Work And Kinetic Energy
Section: Chapter Questions
Problem 108CP: Consider a linear spring, as in Figure 7.7(a), with mass M uniformly distributed along its length....
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mass m is fixed to
Pendulum A simple pendulum consisting of a point
1.
the end of a massless rod (length /), whose other end is pivoted from the ceiling to let
it swing freely in a vertical plane,
specified by its angle o from the equilibrium position.
(a) Prove that the pendulum's potential energy (measured from the equilibrium level) is
as shown below. The pendulum's position can be
U ()mgl (1 - cos )
(b) Write down the total energy E as a function of and o. Show that by differentiating
your expression for E with respect to t you can get the equation of motion for ¢ and
that the equation of motion is just the familiar T
moment of inertia, and a is the angular acceleration ).
(c) Assuming that the angle remains small throughout the motion, show that the motion
is periodic with period
Ia (where T is the torque, I is the
- 2т V1/9.
Tо
m
1
Transcribed Image Text:mass m is fixed to Pendulum A simple pendulum consisting of a point 1. the end of a massless rod (length /), whose other end is pivoted from the ceiling to let it swing freely in a vertical plane, specified by its angle o from the equilibrium position. (a) Prove that the pendulum's potential energy (measured from the equilibrium level) is as shown below. The pendulum's position can be U ()mgl (1 - cos ) (b) Write down the total energy E as a function of and o. Show that by differentiating your expression for E with respect to t you can get the equation of motion for ¢ and that the equation of motion is just the familiar T moment of inertia, and a is the angular acceleration ). (c) Assuming that the angle remains small throughout the motion, show that the motion is periodic with period Ia (where T is the torque, I is the - 2т V1/9. Tо m 1
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