Goal Apply Pascal\'s principle to a car lift, and show that the input work is th

Question

Goal Apply Pascal's principle to a car lift, and show that the input work is the same as the output work.

Problem In a car lift used in a service station, compressed air exerts a force on a small piston of circular cross section having a radius of r1 = 4.95 cm. This pressure is transmitted by an incompressible liquid to a second piston of radius 15.8 cm.

(a) What force must the compressed air exert on the small piston in order to lift a car weighing 13,300 N? Neglect the weights of the pistons.

(b) What air pressure will produce a force of that magnitude?

(c) Show that the work done by the input and output pistons is the same.

Strategy Substitute into Pascal's principle in part (a), while recognizing that the magnitude of the output force, F2, must be equal to the car's weight in order to support it. Use the definition of pressure in part (b). In part (c), use W = Fx to find the ratio W1/W2, showing that it must equal 1. This requires combining Pascal's principle with the fact that the input and output pistons move through the same volume. Solution (a) Find the necessary force on the small piston. Substitute known values into Pascal's principle, using A = r2 for the area.

F1 = N (b) Find the air pressure producing F1. Substitute into the definition of pressure.
P = Pa (c) Show that the work done by the input and output pistons is the same. First equate the volumes, and solve for the ratio of A2 to A1. V1 = V2A1x1 = A2x2
Now use Pascal's principle to get a relationship for F1/F2. Evaluate the work ratio, substituting the preceding two results.
Thus: W1 = W2
Remarks In this problem, we didn't address the effect of possible differences in the heights of the pistons. If the column of fluid is higher in the small piston, the fluid weight assists in supporting the car, reducing the necessary applied force. If the column of fluid is higher in the large piston, both the car and the extra fluid must be supported, so additional applied force is required.

Explanation / Answer

(A) P = F1 / A1 = F2 / A2 = constant

13,300 / (pi (15.8^2)) = F / (pi (4.95^2))

F = 1305.4 N

(b) P = 13,300 N / (pi (0.158^2))

P = 169585 Pa

(c) Answer will be same as given in solution.

(as we have to prove only. No numerical value)

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