- #1
Tanya Sharma
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Homework Statement
A spherical ball of mass M and radius r rolls without slipping in a cylindrical of radius R .Find the time period of small oscillations .
Homework Equations
The Attempt at a Solution
Please have a look at the picture attached .
The ball moves to the right and upward, and rotates about its centre clockwise . ##\phi## is the angle from the vertical by which the ball has turned .##\theta## is the angular displacement of the CM of the ball.
##\ddot{\theta}## is the angular acceleration of the CM of the ball about the center of the curve .
##\ddot{\phi}## is the angular acceleration of the ball about its CM .
Considering anticlockwise as positive and rightwards as positive .
Equation for the motion of the CM :
##f_s - Mgsin\theta = Ma##
Equation for the rotation about the CM
##f_sr = -I\ddot{\phi}##
Rolling without slipping condition :
##a=\ddot{\phi}r##
Constraint equation :
##\ddot{\phi}r = (R-r)\ddot{\theta}##
Solving the above gives the correct time period .
Ok till here but if I consider the ball rolling down from right towards A , I get confused with the signs in the first two equations.
1) Even though I have written the correct torque equation ##f_sr = -I\ddot{\phi}## , I am not quite confident with the reasoning I had regarding the negative sign on the RHS . Is it because ##\phi## is increasing clockwise and I have taken anticlockwise as positive Or is it because ##\ddot{\phi}## is decelerating the rotation ?
2) How do I determine the sign of ##\ddot{\phi}## and ##\ddot{\theta}## ?
3) How should I modify the first two equations ?
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