Question

An object oscillates back and forth on the end of a spring. Which of the following statements are true at some time during the course of the motion?a. The object can have zero acceleration and, simultaneously, nonzero velocity.b. The object can have zero velocity and, simultaneously, zero acceleration.c. The object can have nonzero velocity and nonzero acceleration simultaneously.d. The object can have zero velocity and, simultaneously, nonzero acceleration.

Answer 1

Answer:

a. The object can have zero acceleration and, simultaneously, nonzero velocity.

c. The object can have nonzero velocity and nonzero acceleration simultaneously.

d. The object can have zero velocity and, simultaneously, nonzero acceleration.

Explanation:

For an object in simple harmonic motion, the total mechanical energy (sum of elastic potential energy and kinetic energy) is constant:

$E=U+K=\frac{1}{2}kx^2 + \frac{1}{2}mv^2$ (1)

where

k is the spring constant

x is the displacement

m is the mass

v is the speed

We can also notice that the force on the spring is given by Hook's law:

$F=-kx$

And since according to Newton's law we have F = ma, this can be rewritten as

$ma=-kx\\a=-\frac{k}{m}x$

which means that the acceleration is proportional to the displacement.

So by looking again at eq.(1), we can now states that:

- when the displacement is zero, x=0, the acceleration is zero, a=0, and the velocity is maximum

- when the velocity is zero, v=0, the acceleration is maximum, which occurs when the displacement is maximum

- in all the other intermediate situations, both velocity and acceleration are non-zero

So the correct answers are

a. The object can have zero acceleration and, simultaneously, nonzero velocity.

c. The object can have nonzero velocity and nonzero acceleration simultaneously.

d. The object can have zero velocity and, simultaneously, nonzero acceleration.