The surface area of a small piston is $\frac{1}{10}$ the surface area of a large piston. Find the input force if the output force is $20,000 N.
Answer (2)
Define variables for the surface areas and forces on the small and large pistons.
Use the given information to establish the relationship between the surface areas: A s = 10 1 A l .
Apply Pascal's principle: A s F s = A l F l .
Calculate the input force: F s = F l ⋅ A l A s = 20 , 000 N ⋅ 10 1 = 2000 N .
Explanation
Problem Analysis We are given that the surface area of the small piston is 10 1 the surface area of the large piston. We are also given that the output force is 20 , 000 N . We need to find the input force.
Define variables Let A s be the surface area of the small piston and A l be the surface area of the large piston. Let F s be the input force (force on the small piston) and F l be the output force (force on the large piston).
State given information We are given that A s = 10 1 A l , which can be rewritten as A l A s = 10 1 . We are given that F l = 20 , 000 N .
Apply Pascal's Principle According to Pascal's principle, the pressure is the same throughout the fluid, so A s F s = A l F l .
Rearrange the formula Therefore, F s = F l ⋅ A l A s .
Substitute the values Substitute the given values: F s = 20 , 000 N ⋅ 10 1 .
Calculate the input force Calculate F s = 10 20000 = 2000 N .
Final Answer The input force is 2000 N .
Examples
Hydraulic systems are used in many applications, such as car brakes and lifts. The principle behind these systems is Pascal's principle, which states that pressure is transmitted equally throughout a fluid. This means that a small force applied to a small area can be multiplied to produce a large force on a large area. For example, if you apply a force of 100 N to a small piston with an area of 1 square centimeter, you can produce a force of 1000 N on a large piston with an area of 10 square centimeters. This principle is used in car brakes to allow you to stop a heavy car with a relatively small force on the brake pedal.
The input force needed for the small piston is 2 , 000 N when the output force from the large piston is 20 , 000 N . This is calculated by using the ratio of the surface areas between the two pistons and applying Pascal's principle. Hence, using F s = F l ⋅ A l A s gives us the result.
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