The diagram shows a small body on the inside of a smooth half cylinder of radius R. It is released on the surface at an initial angle of 0. Assume that is small enough for sin 0, 0, to be valid. R (a) Prove that the body executes SHM and find the period. (b) Calculate the time at which the body passes the lowest point for the first time, and its velocity there. (c) Use energy methods to compute that velocity and explain the difference between the two results. Hint: exploit the similarity of this system to the mathematical pendulum and use the approximation cose 1-2/²

The diagram shows a small body on the inside of a smooth half cylinder of radius R. It is released on the surface at an initial angle of 0. Assume that is small enough for sin 0, 0, to be valid. R (a) Prove that the body executes SHM and find the period. (b) Calculate the time at which the body passes the lowest point for the first time, and its velocity there. (c) Use energy methods to compute that velocity and explain the difference between the two results. Hint: exploit the similarity of this system to the mathematical pendulum and use the approximation cose 1-2/²

Classical Dynamics of Particles and Systems

5th Edition

ISBN:9780534408961

Author:Stephen T. Thornton, Jerry B. Marion

Publisher:Stephen T. Thornton, Jerry B. Marion

Chapter12: Coupled Oscillations

Section: Chapter Questions

Problem 12.4P: Refer to the problem of the two coupled oscillators discussed in Section 12.2. Show that the total...

See similar textbooks

Similar questions

Check Your Understanding Identify an object that undergoes uniform circular motion. Describe how you could trace the SHM of this object.
Some people modify cars to be much closer to the ground than when manufactured. Should they install stiffer springs? Explain your answer.
Show that, if a driven oscillator is only lightly damped and driven near resonance, the Q of the system is approximately Q2(TotalenergyEnergylossduringoneperiod)
Refer to the problem of the two coupled oscillators discussed in Section 12.2. Show that the total energy of the system is constant. (Calculate the kinetic energy of each of the particles and the potential energy stored in each of the three springs, and sum the results.) Notice that the kinetic and potential energy terms that have 12 as a coefficient depend on C1 and 2 but not on C2 or 2. Why is such a result to be expected?
Check Your Understanding Identify one way you could decrease the maximum velocity of a simple harmonic oscillator.
Show that the time rate of change of mechanical energy for a damped, undriven oscillator is given by dE/dt = bv2 and hence is always negative. To do so, differentiate the expression for the mechanical energy of an oscillator, E=12mv2+12kx2, and use Equation 12.28.
Give an example of a simple harmonic oscillator, specifically noting how its frequency is independent of amplitude.
Consider a damped harmonic oscillator. After four cycles the amplitude of the oscillator has dropped to 1/e of its initial value. Find the ratio of the frequency of the damped oscillator to its natural frequency.
If a car has a suspension system with a force constant of 5.00104 N/m , how much energy must the car’s shocks remove to dampen an oscillation starting with a maximum displacement of 0.0750 m?
Most harmonic oscillators are damped and, if undriven, eventually come to a stop. Why?
Find the frequency of a tuning fork that takes 2.50103 s to complete one oscillation.
Calculate the maximum values of the amplitudes of the response functions shown in Figures 3-22 and 3-24. Obtain numerical values for β = 0.2ω0 when a = 2 m/s2, ω0 = 1 rad/s, and t0 = 0.