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

[Ayesha02]

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=coefficient of friction of surface but I am unable to plot the graph post that time

 

[ehild]

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=coefficient 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.

 

[Ayesha02]

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)

 

[ehild]

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?

 

[Ayesha02]

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

 

[ehild]

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?

 

[Ayesha02]

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