Question

Match the following and choose the correct option

Column I(initially)

Column II(when rolling withoutsliping begins)

i.		a	$V_{CM}$ is towards left in case of uniform ring
ii.		b	$V_{CM}$ is towards left in case of solid uniform sphere
iii.		c	$V_{CM}$ is towards right in case of uniform ring
iv.		d	$V_{CM}$ is towards right in case of solid uniform sphere

• A. i$\to$a,b,ii$\to$b,c,iii$\to$a,b,iv$\to$a,b
• B. i$\to$a,d,ii$\to$b,c,iii$\to$a,b,iv$\to$a,b
• C. i$\to$a,b,ii$\to$c,d,iii$\to$a,d,iv$\to$a,b
• D. i$\to$a,d,ii$\to$b,c,iii$\to$a,d,iv$\to$a,d

Answer 1

The correct option is A i$\to$a,b,ii$\to$b,c,iii$\to$a,b,iv$\to$a,b

From the options we can see that the object can either be a ring or a sphere. Let us assume the anticlockwise direction as positive. We need to find the direction of center of mass once the body attains pure rolling
In case (i), for ring,
According to the assumption $\omega$ is positive.
$L_i=M{R}^2\omega +MVR=3MVR$ $\because \omega =\frac{2V}{R}\Rightarrow Positive$
So we can conclude that the center of mass will continue to move towads left.
For sphere,
$L_i=\frac{2}{5}M{R}^2\omega +MVR$
$=\frac{9}{5}MVR\Rightarrow Positive$
Again since angular momentum is positive, the body will continue to move towards left.

In Case (ii) In this case $\omega$ is clockwise
$\therefore \omega =-\frac{2v}{R}\Rightarrow Negative$
for ring,
$L_i=-MVR\Rightarrow Negative$
Therefore the center of mass is moving towards right.
For sphere, $L_i=\frac{1}{5}MVR\Rightarrow$
So, the center of mass moves towards left.

In case (iii), $\omega =\frac{2V}{R}$ as it is in anticlockwise direction
for ring, $L_i=\frac{3}{2}MVR\Rightarrow Positive$
So, the center of mass moves towards left.
For sphere, $L_i=\frac{6}{5}MVR\Rightarrow Positive$
So, the center of mass moves towards left.


In case (iv), $\omega$ is clockwise
$\therefore \omega =-\frac{2v}{R}\Rightarrow Negative$
For ring, $L_i=\frac{MVR}{2}\Rightarrow Positive$
So, the center of mass moves towards left.

For sphere, $L_i\Rightarrow Positive$
So, the center of mass moves towards left.