Rotational Dynamics of a Cylinder Rotating Under an Applied Force

In summary, the problem involves a light rigid cylinder with a particle attached to it at a distance a from the axis. A light string is wound around the cylinder and a tension F is applied to it. The goal is to find the angular acceleration and velocity of the cylinder after it has rotated through an angle θ and to determine a limiting tension F0. The equations for angular velocity and acceleration are ω2 = (4Fθ/(ma)) -2g(1-cos(θ))/a and α=(2F/ma)-(g*sin(θ))/a. A previous approach of setting ω=0 at θ=π and solving for F yielded a value of F=mg/π, which does not match
  • #1
MissMoneypenny
17
0

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=[itex]\pi[/itex] 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 [itex]\pi[/itex]. 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 :)
 
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  • #2
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.)
 
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  • #3
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
 
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1. What is rotational dynamics?

Rotational dynamics is the branch of physics that studies the movement and behavior of objects that are rotating under the influence of an external force.

2. How does a cylinder rotate under an applied force?

A cylinder will rotate under an applied force according to the principles of Newton's laws of motion. The direction and magnitude of the rotation will depend on the direction and magnitude of the applied force, as well as the mass and shape of the cylinder.

3. What is the relationship between torque and rotational dynamics?

Torque is the measure of the tendency of a force to cause rotation. In rotational dynamics, torque is a crucial concept as it determines the rate at which a cylinder will rotate under an applied force. The higher the torque, the greater the rotational acceleration of the cylinder.

4. What factors can affect the rotational dynamics of a cylinder?

The rotational dynamics of a cylinder can be affected by various factors, including the magnitude and direction of the applied force, the mass and shape of the cylinder, and the surface on which it is rotating. Friction and air resistance can also play a role in the rotational dynamics of a cylinder.

5. How is the rotational motion of a cylinder calculated?

The rotational motion of a cylinder can be calculated using the principles of rotational dynamics. The moment of inertia, which is a measure of the object's resistance to rotational motion, and the applied torque are key factors in the calculation. The equations of rotational motion can also be used to determine the angular velocity and acceleration of the cylinder.

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