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7.43 *** (Computer] Consider a massless wheel of radius R mounted on a frictionless horizontal axis. A point máss M is glued to the edge, and a massless string is wrapped several times around the perimeter and hangs vertically down with a mass m suspended from its bottom end. (See Figure 4.28.) Initially I am holding the wheel with M vertically below the axle. At 1 = 0, I release the wheel, and m starts to fall vertically down. (a) Write down the Lagrangian L = T - U as a function of the angle o through which the wheel has turned. Find the equation of motion and show that, provided m < M, there is one position of stable equilibrium. (b) Assuming m < M, sketch the potential energy U () for -n < ¢ < 47 and use your graph to explain the equilibrium position you found. (c) Because the equation of motion cannot be solved in terms of elementary functions, you are going to solve it numerically. This requires that you choose numerical values for the various parameters. Take M = g = R = 1 (this amounts to a convenient choice of units) and m = 0.7. Before solving the equation make a careful plot of U ($) against o and predict the kind of motion expected when M is released from rest at ø = 0. Now solve the equation of motion for 0 < t < 20 and verify your prediction. (d) Repeat part (c), but with m = 0.8.

7.43 *** (Computer] Consider a massless wheel of radius R mounted on a frictionless horizontal axis. A point máss M is glued to the edge, and a massless string is wrapped several times around the perimeter and hangs vertically down with a mass m suspended from its bottom end. (See Figure 4.28.) Initially I am holding the wheel with M vertically below the axle. At 1 = 0, I release the wheel, and m starts to fall vertically down. (a) Write down the Lagrangian L = T - U as a function of the angle o through which the wheel has turned. Find the equation of motion and show that, provided m < M, there is one position of stable equilibrium. (b) Assuming m < M, sketch the potential energy U () for -n < ¢ < 47 and use your graph to explain the equilibrium position you found. (c) Because the equation of motion cannot be solved in terms of elementary functions, you are going to solve it numerically. This requires that you choose numerical values for the various parameters. Take M = g = R = 1 (this amounts to a convenient choice of units) and m = 0.7. Before solving the equation make a careful plot of U ($) against o and predict the kind of motion expected when M is released from rest at ø = 0. Now solve the equation of motion for 0 < t < 20 and verify your prediction. (d) Repeat part (c), but with m = 0.8.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Classical Mechanics by John R Taylor.

Subparts (a), (b) and (c) are required.

 

R M Figure 4.28
Transcribed Image Text:R M Figure 4.28
7.43 *** [Computer] Consider a massless wheel of radius R mounted on a frictionless horizontal axis. A point máss M is glued to the edge, and a massless string is wrapped several times around the perimeter and hangs vertically down with a mass m suspended from its bottom end. (See Figure 4.28.) Initially I am holding the wheel with M vertically below the axle. Ati = 0, I release the wheel, and m starts to fall vertically down. (a) Write down the Lagrangian L = T - U as a function of the angle ø through which the wheel has turned. Find the equation of motion and show that, provided m < M, there is one position of stable equilibrium. (b) Assuming m < M, sketch the potential energy U (4) for --n <¢ < 4n and use your graph to explain the equilibrium position you found. (c) Because the equation of motion cannot be solved in terms of elementary functions, you are going to solve it numerically. This requires that you choose numerical values for the various parameters. Take M = g = R = 1 (this amounts to a convenient choice of units) and m = 0.7. Before solving the equation make a careful plot of U ($) against o and predict the kind of motion expected when M is released from rest at ø = 0. Now solve the equation of motion for 0 <t < 20 and verify your prediction. (d) Repeat part (c), but with m = 0.8.
Transcribed Image Text:7.43 *** [Computer] Consider a massless wheel of radius R mounted on a frictionless horizontal axis. A point máss M is glued to the edge, and a massless string is wrapped several times around the perimeter and hangs vertically down with a mass m suspended from its bottom end. (See Figure 4.28.) Initially I am holding the wheel with M vertically below the axle. Ati = 0, I release the wheel, and m starts to fall vertically down. (a) Write down the Lagrangian L = T - U as a function of the angle ø through which the wheel has turned. Find the equation of motion and show that, provided m < M, there is one position of stable equilibrium. (b) Assuming m < M, sketch the potential energy U (4) for --n <¢ < 4n and use your graph to explain the equilibrium position you found. (c) Because the equation of motion cannot be solved in terms of elementary functions, you are going to solve it numerically. This requires that you choose numerical values for the various parameters. Take M = g = R = 1 (this amounts to a convenient choice of units) and m = 0.7. Before solving the equation make a careful plot of U ($) against o and predict the kind of motion expected when M is released from rest at ø = 0. Now solve the equation of motion for 0 <t < 20 and verify your prediction. (d) Repeat part (c), but with m = 0.8.
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