Question

A small roller of diameter $20cm$ has an axle of diameter $10cm$ (see figure below on the left). It is on a horizontal floor and a meter scale is positioned horizontally on its axle with one edge of the scale on top of the axle (see figure on the right). The scale is now pushed slowly on the axle so that it moves without slipping on the axle, and the roller starts rolling without slipping. After the roller has moved$50cm$, the position of the scale will look like (figures are schematic and not drawn to scale)

• A.

  

• B.

  

• C.

  

• D.

  

Answer 1

The correct option is B

Step 1. Given data:

Diameter of small roller, $d_1=20cm$ $\Rightarrow R=10cm$

Diameter of axle, $d_2=10cm$

Distance roller moved, $x=50cm$

Step 2. Finding the position of scale:

From diagram,

Velocity at the lowest point( on the ground) $v-\omega R=0$

where $v$is the velocity of the center of mass, $\omega$ is the angular velocity, and $R$ is the radius of the roller

$v=\omega R=10\omega$

$\omega =\frac{v}{10}$

Velocity at point $A$ , $v+\omega r$,

Putting the values, we get $v+\frac{v}{10}5=\frac{3v}{2}$

So, the distance moved by scale is $\frac{3v\times time}{2}=\frac{3vt}{2}$

since, $vt=50cm$, the distance moved by scale is $\frac{3\times 50}{2}=75cm$

Hence, option B is correct