Which one of the following statements is true concerning an object executing simple harmonic motion?
a) the objects velocity is never zero
b) the objects acceleration in never zero
c) the objects velocity and acceleration are simultaneously zero
d) the objects velocity is zero when its acceleration is a maximum
e) the objects maximum acceleration is equal to its maximum velocity

D) The objects velocity is zero when its acceleration is a maximum

Explanation:

In a simple harmonic motion, the total energy is constant (if we neglect air resistance and friction), and it is equal to the sum of the elastic potential energy U and the kinetic energy K:

(1)

where

m is the mass

v is the velocity

k is the spring constant

x is the displacement

As a consequence, since E must remain constant, when K increases U decreases, and vice-versa.

Also, in a simple harmonic motion the acceleration of the system is proportional to the negative of the displacement:

(2)

So, combining (1) with (2), we have the following situations:

- When the displacement is zero (x=0), the acceleration is also zero (a=0), and so the velocity is maximum, because the kinetic energy is maximum

- When the displacement is maximum (x=max), the acceleration is also maximum, while the velocity is zero because the kinetic energy is zero

So, the correct statement is

D) The objects velocity is zero when its acceleration is a maximum