Question

In a car lift, compressed air exerts a force \[{F_1}\] on a small piston having a radius of \[5cm\] . This pressure is transmitted to the second piston of a radius \[15cm\] . If the mass of the car to be lifted is \[1350\;kg\] . What is \[{F_1}\]?
\[A)14.7 \times {10^3}N\]
\[B)1.47 \times {10^3}N\]
\[C)2.47 \times {10^3}N\]
\[D)24.7 \times {10^3}N\]

Answer 1

Hint: The pressure is known as the amount of force exerted per unit area. Dimension of pressure is \[M{\text{ }}{L^{ - 1}}\;{T^{ - 2}}\]. SI unit of pressure is \[N/{m^2}\] . The principle was invented by the French scientist called Blaise Pascal. Pascal's law defines that pressure applied to an enclosed fluid will be transmitted without a change in magnitude to every point of the fluid and the walls of the container. According to Pascal's law, a pressure exerted on a piston creates an equal rise in pressure on another piston in a hydraulic system.

Formula used:
$\dfrac{{{F_1}}}{{\;\pi {r_1}^2}} = \dfrac{{{F_2}}}{{\pi {r^2}_2}}$

Complete step by step answer:
Given: ${r_1} = 5cm$, ${r_2} = 5cm$
$F = 1350kg$
From pascal’s law,
${P_1} = {P_2}$
Where ${P_1} = $ Pressure in the piston $1$
${P_2} = $ Pressure in piston $2$
$\dfrac{{{F_1}}}{{{A_1}}} = \dfrac{{{F_2}}}{{{A_2}}}$
${F_1} = $ Force exerted on the piston $1$
${F_2} = $ Force exerted on the piston $2$
Put area formula, $A = \pi {r^2}$
$\dfrac{{{F_1}}}{{\;\pi {r_1}^2}} = \dfrac{{{F_2}}}{{\pi {r^2}_2}}$
We have to find ${F_1}$
$\pi $ is canceled on L.H.S and R.H.S
So, bringing denominator in L.H.S to R.H.S
$\Rightarrow {F_1} = \dfrac{{{F_2}{r_1}^2}}{{{r_2}^2}}$
By applying all Values in the formula,
$\Rightarrow {F_1} = \dfrac{{1350 \times 9.8 \times {{(5 \times {{10}^{ - 2}})}^2}}}{{{{\left( {15 \times {{10}^{ - 2}}} \right)}^2}}}$
$\Rightarrow {F_1} = 1470N$
$\Rightarrow {F_1} = 1.47 \times {10^3}N$

Hence, the correct answer is in option \[(B) \Rightarrow 1.47 \times {10^3}N\].

Note: Pressure can be calculated as
$P = \dfrac{F}{A}$
Where \[P{\text{ }} = \] Pressure
\[F{\text{ }} = \] Force
\[A{\text{ }} = \] Area
Pascal's law is used in our day to day life in a vehicle braking system and new electronic parking system which is applicable in cars are directly moved to the next floor. And it is used in Hydraulic Jacks. It is used in lifting heavy objects. A hydraulic jack consists of two cylinders.