A block of mass 15 kg is placed on a long trolley. The coefficient of static friction between the block and the trolley 0.18. The trolley accelerates from rest with 0.5 m/${{s}^{2}}$ for 20 s and then moves with uniform velocity. Discuss the motion of the block as viewed by (i) stationary observer on the ground (ii) an observer moving with the trolley.

Mass of the block, m = 15 kg

Coefficient of friction between the block and the trolley p, = 0.18

Acceleration of the trolley, a =0.5 m/${{s}^{2}}$ Time, t = 20 s

(i) As block is placed on the trolley, therefore force applied on the block by the trolley

F = ma = 15x0.5 = 7.5N

Force on the block is the reaction applied by trolley on the block, therefore its direction is opposite to the direction of motion of the trolley.

Due to the reaction force applied by the trolley on the block, it tries to move in backward direction but limiting force opposes its motion. If limiting friction is greater than the applied force on the block than it will not move.

Limiting friction force acting on the block (f) = $\mu$ R

= $\mu$ mg = 0.18x15 x9.8 = 26.46 N

As f > F, therefore static friction will adjust itself equal and opposite to the applied force (F). Therefore, block will remain at rest on the trolley.

Hence, the block will appear to be at rest relative to the trolley to a stationary observer on the ground.

After 20 s, trolley moves with uniform velocity, therefore acceleration and hence force applied by the trolley on the block will be zero and hence no force of friction will act on the block. Now, the block will appear to be at rest relative to a stationary observer on the ground.

(ii) An observer moving with the trolley will have an acceleration and therefore the observer will be a non-inertial frame of reference for which law of inertia does not hold good. Therefore, the motion of the block cannot be observed by him and hence, the block will be at rest relative to the observer.