Question

The displacement in simple harmonic motion is a maximum when the 1. velocity is a maximum. 2. kinetic energy is a maximum. 3. velocity is zero. 4. acceleration is zero. 5. linear momentum is a maximum

Answer 1

Answer:

3. velocity is zero.

Explanation:

The velocity of a simple harmonic motion is given by

$v = \omega\sqrt{A^2-x^2}$

Here, ω is the angular velocity, A is the amplitude (or maximum displacement from the equilibrium point) and x is the displacement at any time.

At maximum displacement, x = A. Then

$v = \omega\sqrt{A^2-A^2} = 0$

Therefore, at maximum displacement, velocity is 0.

Practically, this can be observed in a simple pendulum. As it approaches the maximum displacement, its velocity reduces. It becomes zero at this point and then reverses as the pendulum changes course. Then the velocity begins to increase. It becomes maximum at the equilibrium point but once past that, the velocity begins to reduce as it approaches the other amplitude.

For acceleration,

$a = -\omega^2x$

It follows that at maximum displacement, the acceleration is a maximum. The negative sign indicates that it is in an opposite direction to the displacement. Both kinetic energy ($\frac{1}{2}mv^2$) and linear momentum ($mv$) are proportional to velocity; they are therefore both zero at the maximum displacement.