Rotational Mechanics -- A solid sphere is rolled on a rough surface

AI Thread Summary
The discussion centers on the mechanics of a solid sphere rolling on a rough surface, specifically the relationship between linear and angular velocities. The time at which rotation ceases is determined to be 4v_0 / (5 * μ * g), where μ is the coefficient of friction. Participants explore how friction affects both linear and angular velocities, noting that the sphere initially skids before transitioning to rolling. The equations of motion are applied to derive angular acceleration and final angular velocity. The conversation concludes with a confirmation of the principles of angular momentum conservation in the context of rolling motion.
Ayesha02
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Homework Statement
A uniform solid sphere of mass M radius R is placed on a rough surface of given an initial linear velocity ##v_0## and angular velocity ##w_0##, where ##w_0## = ##v_0## /2R. plot the graph of angular speed about the center v/s time t.
Relevant Equations
##T## = I* alpha , where T= torque, alpha= angular acceleration
and
##w_0(f)## = ##w_0(i)##- alpha*t
I found out the time when rotation ceases to be 4 ##v_0## /5*mew*g, where mew=coefficent of friction of surface but I am unable to plot the graph post that time
 
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Ayesha02 said:
Homework Statement:: A uniform solid sphere of mass M radius R is placed on a rough surface of given an initial linear velocity ##v_0## and angular velocity ##w_0##, where ##w_0## = ##v_0## /2R. plot the graph of angular speed about the center v/s time t.
Relevant Equations:: ##T## = I* alpha , where T= torque, alpha= angular acceleration
and
##w_0(f)## = ##w_0(i)##- alpha*t

I found out the time when rotation ceases to be 4 ##v_0## /5*mew*g, where mew=coefficent of friction of surface but I am unable to plot the graph post that time
Is the initial angular velocity lower or greater than the one when the sphere rolls? The friction decreases linear velocity and either increases or decreases angular velocity, till rolling occurs. Determine both as functions of time, Show your work in detail.
 
Last edited:
ehild said:
The friction decreases linear velocity and increases angular velocity. Determine both as functions of time, Show your work in detail.

I just did
##T## = 2/5 M R^2 * α
which gives α =5μg/ 2R

hence
##w_0(f)## = ##w_0(i)## - α t
i.e 0= ##v_0##/2R - 5μg/ 2R *t

hence i got t=4##v_0##/5μg (time when rotation ceases)
 
Ayesha02 said:
I just did
##T## = 2/5 M R^2 * α
which gives α =5μg/ 2R

hence
##w_0(f)## = ##w_0(i)## - α t
i.e 0= ##v_0##/2R - 5μg/ 2R *t

hence i got t=4##v_0##/5μg (time when rotation ceases)
You kick a ball. After a short time, does it skid or does it roll?
 
ehild said:
You kick a ball. After a short time, does it skid or does it roll?

Skid initially, after some time rolls.
Im assuming the surface is rough
 
Ayesha02 said:
Skid initially, after some time rolls.
Im assuming the surface is rough
Yes. And what is the angular speed in case of pure rolling with respect to the linear velocity of the CM?
 
Last edited:
ehild said:
Yes. And what is the angular speed in case of pure rolling with respect to the linear velocity of the CM?

Angular momentum conservation

I got the answer:bow:
 
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