Question

a block of mass ( m ) oscillates on a spring with frequency ( f ). which of the following expressions is equal to the spring constant of the spring?
a ( \frac{2pi m}{f^2} )
b ( \frac{4pi^2 m}{f^2} )
c ( 2pi m f^2 )
d ( 4pi^2 m f^2 )

Explanation:

Step1: Recall the formula for the frequency of a mass - spring system

The frequency \( f \) of a mass - spring system is given by the formula \( f=\frac{1}{2\pi}\sqrt{\frac{k}{m}} \), where \( k \) is the spring constant and \( m \) is the mass of the block.

Step2: Solve the formula for \( k \)

First, square both sides of the equation \( f = \frac{1}{2\pi}\sqrt{\frac{k}{m}} \) to get rid of the square root. We have \( f^{2}=\frac{1}{4\pi^{2}}\cdot\frac{k}{m} \).
Then, multiply both sides of the equation by \( 4\pi^{2}m \) to isolate \( k \). So, \( k = 4\pi^{2}m f^{2} \).

Answer:

D. \( 4\pi^{2}mf^{2} \)