A block with a mass of $4\ \mathrm{kg}$ is attached to a spring on the wall that oscillates back and forth with a frequency of $4\ \mathrm{Hz}$ and an amplitude of $3\ \mathrm{m}$. What would the frequency be if the block were replaced by one with one‑fourth the mass and the amplitude of the block is increased to $9\ \mathrm{m}$?
A) $4\ \mathrm{Hz}$
B) $8\ \mathrm{Hz}$
C) $12\ \mathrm{Hz}$
D) $24\ \mathrm{Hz}$
A) $4\ \mathrm{Hz}$
B) $8\ \mathrm{Hz}$
C) $12\ \mathrm{Hz}$
D) $24\ \mathrm{Hz}$
The frequency of a spring‑block simple harmonic oscillator is independent of the amplitude. The equation for the frequency of a spring‑block simple harmonic oscillator is
$$f=\frac{1}{2\pi }\sqrt{\frac{k}{m}}$$
The frequency is inversely proportional to the square root of the mass, so decreasing the mass of the block by a factor of $4$ would increase the frequency by a factor of $2$.