Question

An object is in simple harmonic motion with a period of T. If the object was displaced in the positive direction and released at time t = 0, which one of the following statements correctly describes the motion of the object at time t = 3/4 T?

1) The object is momentarily at rest at maximum negative displacement.
2) The object is passing through the equiLiBrium position, moving in the negative direction.
3) The object is passing through the equiLiBrium position, moving in the positive direction.
4) The object is halfway between the equiLiBrium position and the position of maximum negative displacement.
5) The object is halfway between the equiLiBrium position and the position of maximum positive displacement.

Answer 1

At time t = 3/4 T, an object in SHM is at its maximum negative displacement and momentarily at rest, which corresponds to option 1 of the statements provided.

Step-by-step explanation:

An object in simple harmonic motion (SHM) with period T undergoes a complete cycle every T seconds. If released from maximum positive displacement at t = 0, it would reach the equilibrium position at t = T/4, continue to the maximum negative displacement at t = T/2, and then return towards equilibrium, passing through the equilibrium position again at t = 3T/4.

At t = 3/4 T, the object has completed three-quarters of a period. Since it takes half a period (T/2) to go from one extreme of motion to the other, t = 3T/4 represents the moment just before the object reaches the maximum displacement on the next half of the cycle. At this point (3/4 T), the object is therefore momentarily at rest at maximum negative displacement.

Accordingly, the correct description of the object's motion at t = 3/4 T is this: (1) The object is momentarily at rest at maximum negative displacement.