# The scale of a spring balance reading from 0 to 175 N has a length of 14.5 cm. A fish hanging...

## Question:

The scale of a spring balance reading from 0 to 175 N has a length of 14.5 cm. A fish hanging from the bottom of the spring oscillates vertically at a frequency of 3.00 Hz. Ignoring the mass of the spring, calculate the mass of the fish. Leave your answer in kg.

## Spring Constant:

The spring constant is the parameter that is used to calculate the stiffness of the spring. The spring constant is also known as the force constant of the spring. The spring constant is measured in terms of the Newton per meter.

## Answer and Explanation: 1

**Given Data**

- The maximum force measured by the spring balance is {eq}F = 175\,{\rm{N}} {/eq}.

- The length of the spring balance is {eq}l = 14.5\,{\rm{cm}} {/eq}.

- The frequency of the oscillation is {eq}f = 3.00\;{\rm{Hz}} {/eq}.

The expression for the force constant of the spring is,

{eq}k = \dfrac{F}{l} {/eq}

Substitute the given values in the above expression.

{eq}\begin{align*} k &= \dfrac{{175\;{\rm{N}}}}{{14.5\;{\rm{cm}}\left( {\dfrac{{1\;{\rm{m}}}}{{100\;{\rm{cm}}}}} \right)}}\\[0.3 cm] &= \dfrac{{175\;{\rm{N}}}}{{0.145\;{\rm{m}}}}\\[0.3 cm] &= {\rm{1206}}{\rm{.89}}\;{\rm{N/m}} \end{align*} {/eq}

The expression for the mass of the fish is,

{eq}\begin{align*} f &= \dfrac{1}{{2\pi }}\sqrt {\dfrac{k}{m}} \\[0.3 cm] {f^2} &= \dfrac{1}{{4{\pi ^2}}}\left( {\dfrac{k}{m}} \right)\\[0.3 cm] m &= \dfrac{1}{{4{\pi ^2}}}\left( {\dfrac{k}{{{f^2}}}} \right) \end{align*} {/eq}

Substitute the given values in the above expression.

{eq}\begin{align*} m &= \dfrac{1}{{4{\pi ^2}}}\left( {\dfrac{{1206.89\;{\rm{N/m}}}}{{{{\left( {3\;{\rm{Hz}}\left( {\dfrac{{1\;{{\rm{s}}^{ - 1}}}}{{1\;{\rm{Hz}}}}} \right)} \right)}^2}}}} \right)\\[0.3 cm] &= {\rm{3}}{\rm{.396}}\;{\rm{kg}}\\[0.3 cm] &{\rm{ = 3}}{\rm{.4}}\;{\rm{kg}} \end{align*} {/eq}

Thus, **the mass of the fish is {eq}\boxed{{\rm{3}}{\rm{.4}}\;{\rm{kg}}}
{/eq}.**

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Chapter 17 / Lesson 11In this lesson, you'll have the chance to practice using the spring constant formula. The lesson includes four problems of medium difficulty involving a variety of real-life applications.