EXAMPLE 9.7 The Car Lift 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 r = 5.00 cm. This pressure is transmitted by an incompressible liquid to a second piston of radius 2 = 15.0 cm. (a) What force must the compressed air exert on the small piston in order to lift a car weighing 13300 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 = FAX to find the ratio W/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 = r for the area of F1 F2 = each piston: (5.00 x 10-2 m) (1.50 x 10 m) 2 (1.33 x 104 N) = 1.48 103 N (B) Find the air pressure producing F. Substitute into the definition of pressure: F1 P = A 1.48 x 103 N (5.00 x 10-2 m) (C) Show that the work done by the input and output pistons is the same. First equate the volumes, and solve for V = V2AAx = AAx2 the ratio of A2 to A: A Ax1 A Ax2 Now use Pascal's principle to get a F1 F2 F1 A1 - relationship for F1/F2 A1 A2 F2 A2 Evaluate the work ratio, substituting the preceding two results: F2Ax2 W = W2 1.88 x 10 Pa ()()-(J)() LEARN MORE 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. QUESTION If the radius of the output piston is doubled, the output force increases by a factor of: PRACTICE IT Use the worked example above to help you solve this 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 r = 5.10 cm. This pressure is transmitted by an incompressible liquid to a second piston of radius 14.9 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. N (b) What air pressure will produce a force of that magnitude? Pa (c) Show that the work done by the input and output pistons is the same. (Do this on paper. Your instructor may ask you to turn in this work.) instructor may ask you to turn in this work.) EXERCISE HINTS: GETTING STARTED | I'M STUCK! A hydraulic lift has pistons with diameters 7.75 cm and 36.3 cm, respectively. If a force of 825 N is exerted at the input (smaller) piston, what maximum mass can be lifted at the output piston? m = kg