Question

part a

a mass on the end of a spring undergoes simple harmonic motion. at the instant when the mass is at its maximum displacement from equilibrium, what is its instantaneous velocity?

• at maximum displacement, its instantaneous velocity is also at maximum.
• instantaneous velocity cannot be determined without additional information.
• at maximum displacement, its instantaneous velocity is less than its maximum but not zero.
• at maximum displacement, its instantaneous velocity is zero.

Explanation:

Analyze the energy at maximum displacement

Using the Conservation of Energy knowledge point

$$ E = \frac{1}{2}kA^2 = \frac{1}{2}kx^2 + \frac{1}{2}mv^2 $$

At maximum displacement, \(x = \pm A\):

$$ \frac{1}{2}kA^2 = \frac{1}{2}kA^2 + \frac{1}{2}mv^2 \implies \frac{1}{2}mv^2 = 0 \implies v = 0 $$

Determine the instantaneous velocity

Using the Simple Harmonic Motion Speed knowledge point

$$ v(t) = \pm \omega \sqrt{A^2 - x^2} $$

When \(x = \pm A\):

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

Thus, at maximum displacement, the instantaneous velocity of the mass is zero.

Answer:

• At maximum displacement, its instantaneous velocity is also at maximum.
• Instantaneous velocity cannot be determined without additional information.
• At maximum displacement, its instantaneous velocity is less than its maximum but not zero.
• At maximum displacement, its instantaneous velocity is zero. (Correct answer)