Question

Two syringes of different cross sections (without needle) filled with water are connected with a tightly fitted rubber tube filled with water. Diameters of the smaller and larger piston are $1\,cm{\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} 3\,cm$ respectively. If a force of $10\,N$ is applied to the smaller piston then the force exerted on the larger piston is:
A. $30N$
B. $60N$
C. $90N$
D. $100N$

Answer 1

Hint:In order to solve this question, we should know that, pressure is defined as the force acting on a surface per unit of area and in given information, pressure transmitted from one syringe to another will be same as they are connected through rubber tube filled with water so water is uniformly spread, so we will equalize the pressure at both piston ends and then will figure out the force exerted on larger piston.

Complete step by step answer:
According to the question, we have given that Diameters of both smaller and larger piston are as,
${d_{small}} = 1cm = 0.01m$
$\Rightarrow {d_{l\arg e}} = 3cm = 0.03m$
and area of each cross- sectional area is calculated as,
${A_{small}} = \pi \dfrac{{{d^2}_{small}}}{4}$
$\Rightarrow {A_{l\arg e}} = \pi \dfrac{{{d^2}_{l\arg e}}}{4}$
where area of circle is as $A = \pi \dfrac{{{d^2}}}{4}$ where d is diameter and $\pi = 3.14$ constant.

Also force on smaller piston is given as ${F_{small}} = 10$ and let F be the force exerted on larger piston.Now, pressure P is defined as $P = \dfrac{F}{A}$ since, pressure is equal at both piston ends so, for both piston we have,
$\dfrac{{{F_{small}}}}{{{A_{small}}}} = \dfrac{F}{{{A_{l\arg e}}}}$ on putting the values we get,
$\dfrac{{10 \times {A_{l\arg e}}}}{{{A_{small}}}} = F$
or
$F = \dfrac{{10 \times \pi \times {D^2}_{l\arg e}}}{{\pi \times {D^2}_{small}}}$
$ \Rightarrow F = \dfrac{{10 \times 0.0009}}{{0.0001}}$
$ \therefore F = 90\,N$
So, the force on a larger piston will have a force of $F = 90\,N$.

Hence, the correct option is C.

Note:It should be remembered that, since the whole system is filled with water uniformly so when we exert pressure through one side it will transmit through the system equally and hence pressure at both piston areas is equal and the unit of pressure is known as Pascal.