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Wize University Physics Textbook (Master) > Fluid Mechanics

Hydraulic Systems

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Hydraulic Systems


The Pascal's principle and the incompressibility of liquids make hydraulic systems suitable for manipulating and transmission of forces.


In these hydraulic systems, the volume of fluid moved in the right and left sides are the same


A1d1=A2d2\boxed{A_1d_1=A_2d_2}


where AA is the cross-sectional area of each side and dd is the displacement of each side.




Now imagine pistons of both arms are at the same height. Thus, due to Pascal's principle, the pressure on both sides should be equal. this implies:
F1A1=F2A2\boxed{\dfrac{F_1}{A_1}=\dfrac{F_2}{A_2}}

Exam Tip
Above formula could be used to find the force needed one one piston to keep a load on the other piston at the same height .




Example: Hydraulic Lift


In the hydraulic lift, a large piston supports a car. The radius of the large piston is 240240 cm and that of the smaller piston is 3030 cm. The total mass of car and piston is 32003200 kg.

What force must be applied to the smaller piston to support the car (assuming the two pistons are at to be kept at the same height)?


The pressure applied at the small piston can be used to support the car. Since they are at the same level, each pressure is given by P=FAP=\dfrac{F}{A} . The force on the large piston is the weight of the piston and car:


P1=P2P_1=P_2

F1πr12=F2πr22\dfrac{F_1}{\pi r_1^2}=\dfrac{F_2}{\pi r_2^2}

F1πr12=mgπr22\dfrac{F_1}{\cancel\pi r_1^2}=\dfrac{mg}{\cancel\pi r_2^2}

F1=mg r12r22F_1=\dfrac{mg \ r_1^2}{ r_2^2}


Put the numbers in:

=3200×9.81×0.322.42=\dfrac{3200\times 9.81\times0.3^2}{2.4^2}

=491=491 N

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Example: Lifting a Car by a Hydraulic System


Consider a car of mass mm on a hydraulic lift, as shown. Initially, it is at height hh above the dotted line, which requires applying force FF on the left piston. Then, we want to raise the car by a distance d2d_2. We accomplish this by pushing down on the left side of the tube by a distance d1d_1, with new force F′F'.

a) What was the initial force required to keep the car at a distance hh above the dotted line? (Answer in terms of the areas of the two pistons, mm and hh.)
b) Compared to that point, how much extra force do we need to raise the car by the extra distance d2d_2? (Answer in terms of the areas of the two pistons and the distance d2d_2.)