# A cylinder is fitted with a piston, beneath which is a spring. The cylinder is open at the top....

## Question:

A cylinder is fitted with a piston, beneath which is a spring. The cylinder is open at the top. Friction is absent. The spring constant of the spring is 2500 N/m. The piston has a negligible mass and a radius of 0.0229 m.

(a) When air beneath the piston is completely pumped out, how much does the atmospheric pressure cause the spring to compress?

(b) How much work does the atmospheric pressure do in compressing the spring?

## Stiffness of Spring:

When an external force is applied to the spring, the work done by the force to compress the spring is directly proportional to the stiffness of the spring. The stiffness of the spring is expressed in N/m.

## Answer and Explanation: 1

We are given the following data:

- Spring constant, {eq}k=2500\ \rm N/m {/eq}

- Radius of piston, {eq}r=0.0229\ \rm m {/eq}

#### Question (a)

The force exerted by the atmospheric pressure to compress the spring is expressed by the following equation:

{eq}P\times A=kx {/eq}

Where

- P is the atmospheric pressure
- A is the area of the piston
- x is the compression of the spring

Substituting values in the above equation, we have:

{eq}\begin{align} P\times A&=kx\\[0.3 cm] 101325\ \rm N/m^{2}\times \pi r^{2}&=kx\\[0.3 cm] 101325\ \rm N/m^{2}\times \pi \times\left (0.0229\ \rm m \right )^{2}&=2500\ \rm N/m\times x\\[0.3 cm] x&=\frac{101325\ \rm N/m^{2}\times \pi \times\left (0.0229\ \rm m \right )^{2}}{2500\ \rm N/m}\\[0.3 cm] &=\boxed{0.0667\ \rm m} \end{align} {/eq}

#### Question (b)

The work done by the atmospheric pressure to compress the spring is expressed by the following equation:

{eq}\begin{align} W&=\dfrac{1}{2}kx^{2}\\[0.3 cm] &=\dfrac{1}{2}\times2500\ \rm N/m\times\left ( 0.0667\ \rm m \right )^{2}\\[0.3 cm] &=\boxed{5.57\ \rm N.m}\\ \end{align} {/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.