Question

You stand on the end of a diving board and bounce to set it into oscillation. You find a maximum response in terms of the amplitude of oscillation of the end of the board when you bounce at frequency f. You now move to the middle of the board and repeat the experiment. Is the resonance frequency for forced oscillations at this point (a) higher, (b) lower, or (c) the same as f?

Short answer

The resonance frequency for forced oscillations at this point is higher, hence option (a) is correct.

Step-by-step solution

Simple Harmonic Motion

When an object undergoes simple harmonic motion, frequency of oscillation may be written as

$f=\frac{\omega }{2\pi }=\frac{1}{2\pi }\sqrt{\frac{k}{m}}$

$f=$Frequency of oscillation

$\omega =$Angular frequency

$m=$Mass of object

Step 2: Explain the resonance frequency for forced oscillations

Higher frequency:

• The center of the diving board flexes down less than the end does when it supports our weight this is similar to a spring that stretches a smaller distance for the same force:
• The displacement is smaller, because the spring constant is higher.
• This result into the center of the board is greater than the stiffness constant describing the end.
• $f=\frac{\omega }{2\pi }=\frac{1}{2\pi }\sqrt{\frac{k}{m}}$is greater for we bouncing on the center of the board.

Answer (a) is correct.