The ratio of the diameters of the small piston to the large piston of a hydraulic lift is 1:10. Find (a) the ratio of the force on the small piston to the force on the large piston and (b) the ratio of the pressure on the small piston to the pressure on the big piston.

a)     1: 100  F₂ = 100 F₁,  b) P₁ = P₂   1:1

Explanation:

The pressure in a system is the same for a given horizontal line, therefore

          P₁ = P₂

where subscript 1 is for small piston and index 2 is for large piston 2

a) they ask the relation of the forces

pressure is defined by

        P = F / A

for the small piston

        P₁ = F₁ / A₁

the area of ​​the piston is

        A₁ =π r₁²

        P₁ = F₁ /π r₁²

for big piston

         P₂ = F₂ / A₂

in the exercise indicates that the diameter of the large piston is 10 times the diameter of the small piston

         A = π r₂²

         r₂ = 10 r₁

          r₂ = π 10² r₁²

we substitute

         P₂ =

we equalize the pressures

           F₁ = F₂ / 100

            F₂ / F₁ = 100

therefore the force ratio is

             1: 100

b) according to Pascal's principle for the same height the pressure is the same regardless of the shape of the container, therefore the pressure of the two pistons is the same

           P1 = P2

             1: 1

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