Question

In Figure, a spring of spring constant $\mathbf{3}\mathbf{.00}\mathbf{\times }{\mathbf{10}}^{\mathbf{4}}\mathbf{N}\mathbf{/}\mathbf{m}$is between a rigid beam and the output piston of a hydraulic lever. An empty container with negligible mass sits on the input piston. The input piston has area $\mathbf{}{\mathbf{A}}_{\mathbf{1}}$, and the output piston has area $\mathbf{18}\mathbf{.}\mathbf{0}\mathbf{}{\mathbf{A}}_{\mathbf{1}}$. Initially the spring is at its rest length. How many kilograms of sand must be (slowly) poured into the container to compress the spring by $\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{cm}$?

Short answer

Mass of sand required is$8.50\mathrm{kg}$

Step-by-step solution

The given data

(i)Spring Constant,$\mathrm{k}=3\times {10}^4\mathrm{N}/\mathrm{m}$

(ii) Spring compression,$\mathrm{x}=5\mathrm{cm}\mathrm{or}0.05\mathrm{m}$

(iii) Input area of the piston is$\mathrm{A}$

(iv) Output area of the piston is$18\mathrm{A}$

Understanding the concept of Pascal’s law

We can use Pascal’s law to find the force using the area of the piston and the compression of the spring. This force must be equivalent to the weight of the sand. So, using this value of force, we can find the mass of the sand.

Formula:

Relation of the pressure of larger and smaller the piston using Pascal’s law,

$\frac{{\mathrm{F}}_1}{{\mathrm{A}}_1}=\frac{{\mathrm{F}}_2}{{\mathrm{A}}_2}$ (i)

Where,

${\mathrm{F}}_1$and ${\mathrm{F}}_2$are the force and area of sand respectively; ${\mathrm{A}}_1$and ${\mathrm{A}}_2$are the force and area of the container respectively.

Weight of a body, $\mathrm{F}=\mathrm{mg}$(ii)

Force on a spring, $\mathrm{F}=\mathrm{kx}$(iii)

Calculation of mass of sand

${\mathrm{F}}_1$and${\mathrm{A}}_1$ are the force and area of sand respectively;${\mathrm{F}}_2$ and ${\mathrm{A}}_2$are the force and area of the container respectively.

From equation (ii), the weight of sand(in terms of force due to gravity) can be

${\mathrm{F}}_{1=\mathrm{mg}}$

This weight of the sand would cause the pressure, which would be transmitted to the spring resulting in the compression of the spring.

Let the force at the spring be ${\mathrm{F}}_2$. Using equation (iii), we can write ${\mathrm{F}}_2$as

${\mathrm{F}}_2=\mathrm{kx}$

xis the compression of the spring, andkis the spring constant.

Using equation (i) and substituting the given values, we get

$\frac{\mathrm{mg}}{\mathrm{A}}=\frac{\mathrm{kx}}{18\mathrm{A}}$

$\frac{\mathrm{m}\times 9.8}{\mathrm{A}}-\frac{3\times {10}^4\times 0.05}{18\mathrm{A}}$

$\mathrm{m}=8.5\mathrm{kg}$

Hence, the mass of sand is$8.5\mathrm{kg}$