Rotational Dynamics of a Cylinder Rotating Under an Applied Force

[MissMoneypenny]

Homework Statement

A light rigid cylinder of radius 2a is able to rotate freely about its axis, which is horizontal. A particle of mass m is fixed to the cylinder at a distance a from the axis and is initially at rest at its lowest point. A light string is wound on the cylinder, and a steady tension F applied to it. Find the angular acceleration and angular velocity of the cylinder after it has turned through an angle θ. Show that there is a limiting tension F₀ such that if F < F₀ the motion is oscillatory, but if F > F₀ it continues to accelerate. Estimate the value of F₀ by numerical approximation. The answer given in the book for F₀ is 0.362mg

I determined the angular velocity and acceleration equations easily enough (I'll include them in the "relevant equations" section). However, I'm having trouble with the rest of the question, specifically estimating F₀. Any help is appreciated!

This problem comes from Classical Mechanics, 5th Edition by Kibble.

Homework Equations

I'll use α for angular acceleration andω for angular velocity. I determined that the angular velocity and acceleration are given by the following equations (I apologize for the format of the equations, but when I used the equation editor they did not display properly):

ω² = (4Fθ/(ma)) -2g(1-cos(θ))/a

α=(2F/ma)-(g*sin(θ))/a

This problem is in the "Energy and Angular Momentum" chapter of my textbook. Perhaps that helps give some impression of a valid approach to the problem.

The Attempt at a Solution

To prove that the limiting tension F₀ exists, I used the following approach:

Set the equation for ω equal to zero at θ=\pi and solve for F. I figured that the limiting tension would change the direction of the particle's motion at its apex, that is when it has rotated through an angle \pi. However, this yields F=mg/π≈0.318mg, which is different than the answer in the back of the book (which, as stated above, is 0.362mg). I'm not really sure what's wrong with this approach. The problem statement says to "use numerical methods to estimate F₀, and my approach isn't complicated enough to merit the phrase "use numerical methods". At this point I'm completely stuck on how to prove the existence of F₀ and on how to estimate its value. Thanks in advance for any help :)

 

[TSny]

Hello MissMoneypenny. Welcome to PF!

Suppose the particle just comes to rest at θ = $\pi$, as you assumed. Would the net torque at that point be in the right direction to make the particle reverse its motion so that it oscillates?

(Your equations for ω² and α look good.)

 

[MissMoneypenny]

Hi TSny,

Thanks so much for the help! You were completely correct that the torque at θ=π would be in the wrong direction to reverse the direction of the cylinder's rotation. Your hint also managed to give me just enough of a push to see where I went wrong and to determine a new (and physically valid) approach. I just managed to solve the problem, obtaining the answer given in the back of the book! Thanks so much for the advice :)

Regards,

MissMoneypenny