Question

A vertical scale on a spring balance reads from 0 to 200 N. The scale has a length of 10.0 cm from the 0 to 200 N reading. A fish hanging from the bottom of the spring oscillates vertically at a frequency of 2.05 Hz.

Ignoring the mass of the spring, what is the mass m of the fish?

Answer 1

Concepts and reason

The concept of the frequency of the spring oscillations is used here.

Initially, find out the force constant of the spring using the expression of the force constant and then use the value of spring constant in the expression of the frequency of the spring to calculate mass.

Fundamentals

The expression of the force constant is given as follows:

$k = \frac{F}{x}$

Here, k is the force constant, F is the force, and x is the spring length.

The expression of the frequency of the spring is given as follows:

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

Here, k is the force constant, and m is the mass.

The force constant of the spring is given as:

$k = \frac{F}{x}$

Substitute 200 N for F and 10 cm for x in the above equation.

$\begin{array}{c}\\k = \frac{{\left( {200{\rm{ N}}} \right)}}{{10.0{\rm{ cm}}\left( {\frac{{{{10}^{ - 2}}{\rm{ m}}}}{{1.0{\rm{ cm}}}}} \right)}}\\\\ = 2000{\rm{ N/m}}\\\end{array}$

Rearrange the expression of frequency of spring to solve for mass.

$\begin{array}{c}\\f = \frac{1}{{2\pi }}\sqrt {\frac{k}{m}} \\\\{f^2} = \frac{1}{{2\pi }}\left( {\frac{k}{m}} \right)\\\\m = \frac{1}{{4{\pi ^2}}}\frac{k}{{{f^2}}}\\\end{array}$

Substitute 2000 N for k and 2.05 Hz for f in the above expression for mass of the fish.

$\begin{array}{c}\\m = \frac{1}{{4{{\left( {3.14} \right)}^2}}}\frac{{2000{\rm{ N}}}}{{{{\left( {2.05{\rm{ Hz}}} \right)}^2}}}\\\\ = 12.05{\rm{ kg}}\\\end{array}$

Ans:

The mass of the fish is 12.05 kg.