Question

The figure shows three arrangements of a block attached to identical springs that are in their relaxed state when the block is centered as shown. Rank the arrangements according to the magnitude of the net force on the block, largest first, when the block is displaced by distance d (a) to the right and (b) to the left. Rank the arrangements according to the work done on the block by the spring forces, greatest first, when the block is displaced by d (c) to the right and (d) to the left.

JE (1) (2) (3)

Answer 1

The restoring force acting on a block attached to a spring is given as, F=-kr Here, F is the force, k is the spring constant, and x is the displacement of the block or the distance by which the spring compressed or stretched. The negative sign in the above equation implies that the direction of restoring force is opposite to the direction of displacement of oscillating body. (a) In case (1), when the block is displaced to the right, the magnitude of the restoring force acting on the block is, F = kd + kd = 2kd The restoring force due to the two springs acting in the left direction. Therefore, the net force is the sum of the restoring force due to the two springs. (2) In case (2), when the block is displaced to the right, the magnitude of the restoring force acting on the block is, F = kd +kd + kd = 3kd When the block is displaced to the right, the two springs on the left get stretched and the spring on the right get compressed. To return to the original position, all the springs will apply force in the left direction. Thus, the restoring force due to all the springs is acting in the left direction, Therefore, the net force is the sum of the restoring force due to the three springs. (3)

In case (3), when the block is displaced to the right, the magnitude of the restoring force acting on the block is, F = kd + kd +kd +kd = 4kd When the block is displaced to the right, the two springs on the left get stretched and the two springs on the right get compressed. To return to the original position, all the springs will apply force in the left direction. Thus, the restoring force due to all the springs is acting in the left direction, Therefore, the net force is the sum of the restoring force due to the all four springs. Hence, the ranking of the arrangements according to the magnitude of net force is, 3>2>1 (b) When the block is displaced to the right, the magnitude of the net force on the block will remain same, only the direction will change. This time the direction of net force is to the right. Hence, the ranking of the arrangements according to the magnitude of net force is, 3>2>1 Work done on the block is given as, W = kd? Work done in case (1) is, W, = kd? + kd? = kd 2

Work done in case (2) is, ws=ke + ka + kad Work done in case (3) is, 2 = 2kd Hence, the ranking of the arrangements according to the work done on the block is, 3 > 2>1 (d) When the block is displaced to the right, the amount of work done will remain same, only the direction of displacement gets reversed. Hence, the ranking of the arrangements according to the work done on the block is, 3>2>1