Question

JEE ADVANCED PHYSICS QUESTION SEPTEMBER 2020

Answer 1

A small roller of diameter 20 cm has an axle of diameter 10 cm (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 50 cm, the position of the scale will look like (figures are schematic and not drawn to

scale)-

solution : As it has given that , roller starts rolling without slipping.

so, V_center = ωR.....(1) , where R is the radius of Roller and ω

is angular velocity of roller.

so velocity of scale = V_center + ωr, where r is radius of axle.

Given, roller moved 50 cm.

so, V_center × t = 50 cm ......(2)

so distance moved by scale = (V_center + ωr) × t

= {V_center + V_center × r/R ) × t [ from equation (1) . ]

given, R = 20 cm, r = 10 cm

= (V_center + V_center × 10/20) × t

= 3V_center/2 × t

= (3/2) V_center × t

from equation (2)

= 3/2 × 50

= 75 cm

Therefore the scale will like as shown in option (B)

Answer 2

Radius of the roller, R = 10 cm.

Radius of axle, r = 5 cm.

Both the roller and axle should have same angular velocity ω.

Let the roller has moved 50 cm during a time t. Let its speed be V = Rω. This is the linear speed of the meter scale too.

As the system is moved from rest, by first equation of motion,

Vt = 50 cm

Rωt = 50 cm

ωt = 5 rad

The meter scale experiences a combination of linear speed as it's pushed, and speed due to rotation of axle. Let its speed be v.

Let the displacement of the meter scale, wrt the ground, be x. Then,

x = vt

x = (V + rω)t

x = (R + r)ωt

x = (10 + 5)5 cm

x = 75 cm

The displacement of the meter scale wrt the roller will be,

Δx = 75 cm - 50 cm

Δx = 25 cm

This means the end of the meter scale, which was initially in contact with the uppermost point of axle, is finally becoming at a distance 25 cm rightwards from the point.

This is correctly depicted in option (B) among them and so option (B) is the answer.