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2. A simple pendulum consists of a point mass m suspended by a massless string with length. The point mass will oscillate back and forth along a circular are with radius . This is an example of a system that approximately undergoes simple harmonic motion under certain conditions. m a) Determine the angular acceleration of the point mass, which should lead you to a differential equation for the angle 0. b) Now suppose the angle is very small. If the angle is small, sin (if you look at a graph of sin 0, for small , you should see that sin is approximately linear). Using this approximation, show that your answer to part a) leads to simple harmonic motion. Find the period, frequency, and angular frequency of the oscillations. c) For this part, suppose = 0.5 m, m = 2 kg, and at t=0 the mass is at rest at an angle = -0.1 rad (note: a negative angle mean's the mass is to the left of the vertical). What is as a function of time? d) Assuming the oscillations are small, what physical properties affect the period of the system? If you increase the mass m, what happens to the period? If you increase or decrease the length of the string, what happens to the period? What if the pendulum was near the surface of the moon instead of the earth, what would happen to the period? Make sure

2. A simple pendulum consists of a point mass m suspended by a massless string with length. The point mass will oscillate back and forth along a circular are with radius . This is an example of a system that approximately undergoes simple harmonic motion under certain conditions. m a) Determine the angular acceleration of the point mass, which should lead you to a differential equation for the angle 0. b) Now suppose the angle is very small. If the angle is small, sin (if you look at a graph of sin 0, for small , you should see that sin is approximately linear). Using this approximation, show that your answer to part a) leads to simple harmonic motion. Find the period, frequency, and angular frequency of the oscillations. c) For this part, suppose = 0.5 m, m = 2 kg, and at t=0 the mass is at rest at an angle = -0.1 rad (note: a negative angle mean's the mass is to the left of the vertical). What is as a function of time? d) Assuming the oscillations are small, what physical properties affect the period of the system? If you increase the mass m, what happens to the period? If you increase or decrease the length of the string, what happens to the period? What if the pendulum was near the surface of the moon instead of the earth, what would happen to the period? Make sure

Physics for Scientists and Engineers
Physics for Scientists and Engineers
10th Edition
ISBN:9781337553278
Author:Raymond A. Serway, John W. Jewett
Publisher:Raymond A. Serway, John W. Jewett
Chapter15: Oscillatory Motion
Section: Chapter Questions
Problem 44AP
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2. A simple pendulum consists of a point mass m suspended by a massless string with length. The point mass will oscillate back and forth along a circular are with radius . This is an example of a system that approximately undergoes simple harmonic motion under certain conditions. m a) Determine the angular acceleration of the point mass, which should lead you to a differential equation for the angle 0. b) Now suppose the angle is very small. If the angle is small, sin ≈ 0 (if you look at a graph of sin 0, for small 0, you should see that sin is approximately linear). Using this approximation, show that your answer to part a) leads to simple harmonic motion. Find the period, frequency, and angular frequency of the oscillations. c) For this part, suppose l = 0.5 m, m = 2 kg, and at t=0 the mass is at rest at an angle = -0.1 rad (note: a negative angle mean's the mass is to the left of the vertical). What is as a function of time? d) Assuming the oscillations are small, what physical properties affect the period of the system? If you increase the mass m, what happens to the period? If you increase or decrease the length of the string, what happens to the period? What if the pendulum was near the surface of the moon instead of the earth, what would happen to the period? Make sure to briefly explain the physical reasoning behind your conclusions. Don't just look at your equation for the period, think about what is happening physically.
Transcribed Image Text:2. A simple pendulum consists of a point mass m suspended by a massless string with length. The point mass will oscillate back and forth along a circular are with radius . This is an example of a system that approximately undergoes simple harmonic motion under certain conditions. m a) Determine the angular acceleration of the point mass, which should lead you to a differential equation for the angle 0. b) Now suppose the angle is very small. If the angle is small, sin ≈ 0 (if you look at a graph of sin 0, for small 0, you should see that sin is approximately linear). Using this approximation, show that your answer to part a) leads to simple harmonic motion. Find the period, frequency, and angular frequency of the oscillations. c) For this part, suppose l = 0.5 m, m = 2 kg, and at t=0 the mass is at rest at an angle = -0.1 rad (note: a negative angle mean's the mass is to the left of the vertical). What is as a function of time? d) Assuming the oscillations are small, what physical properties affect the period of the system? If you increase the mass m, what happens to the period? If you increase or decrease the length of the string, what happens to the period? What if the pendulum was near the surface of the moon instead of the earth, what would happen to the period? Make sure to briefly explain the physical reasoning behind your conclusions. Don't just look at your equation for the period, think about what is happening physically.
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