- #1
Tanya Sharma
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- 135
Homework Statement
A spherical ball of mass m and radius r rolls without slipping on a rough concave surface of large radius R .It makes small oscillations about the lowest point.Find the time period.
Ans : [itex]2∏\sqrt\frac{7(R-r)}{5g}[/itex]
Homework Equations
The Attempt at a Solution
We displace the spherical ball towards right so that it rolls down towards the bottommost position .
Let the origin be fixed at the CM of the ball and fix coordinate axis with + x-axis rightwards upwards
Let θ be the angle which the CM of the ball makes with the center of the concave surface
Let ∅ be the angle which the ball rotates around its axis i.e CM
Let α be the angular acceleration of the CM of the ball around the center of the concave surface
Let β be the angular acceleration of the ball around the CM
For translation motion of the CM of the spherical ball
f - mgsinθ =ma where a is the acceleration of the CM of the ball
fr = Iβ where I=(2/5)mr2
a=βr (no slip condition)
We also have ,
∅=[itex]\frac{R-r}{r}θ[/itex]
So ,β=[itex]\frac{R-r}{r}α[/itex]
Solving above equations ,I get incorrect answer .
I feel the problem is somewhere with signs .If ,we consider frictional torque around CM as -fr instead as fr or β=-([itex]\frac{R-r}{r})α[/itex] ,we get correct answer .
How do assign signs to the torque and the angular accelerations just like we assign positive and negative signs to forces and accelerations ?
Kindly help me with the problem .