A hydraulic lift consists of a platform that is supported by a cylindrical piston, which resides in a tube connected to a reservoir of hydraulic fluid. The piston and platform have a mass of 300 kg. The input force is generated at a smaller piston connected to the same reservoir. If the radius of the smaller piston is 2 cm, what must be the radius of the larger piston so that a moderate force of 100 N will lift a 1,700 kg car?
A 1700 kg car weighs M g = 16,660 Newtons. The piston and platform add 2940 N. That makes 19600 N on the heavy end.
The area ratio of the pistons must be 19,600/10 = 1960
The square root of that number is the ratio of radii.
where did the 10 come from?
To solve this problem, we need to apply Pascal's law, which states that the pressure exerted on a fluid in a confined space is transmitted equally in all directions. In other words, the pressure on the smaller piston will be the same as the pressure on the larger piston.
First, let's find the pressure generated by the force of 100 N on the smaller piston.
Step 1: Convert the radius of the smaller piston to meters.
Radius of smaller piston = 2 cm = 0.02 m
Step 2: Calculate the area of the smaller piston.
Area of smaller piston = π * (radius of smaller piston)^2
Step 3: Calculate the pressure on the smaller piston.
Pressure = Force / Area of smaller piston
Let's calculate it:
Pressure = 100 N / (π * (0.02 m)^2)
Pressure ≈ 79577.47 Pascal (Pa)
Now we need to find the radius of the larger piston which will produce the same pressure. We know that the pressure on the larger piston should also be 79577.47 Pa.
Step 4: Calculate the area of the larger piston.
Area of larger piston = π * (radius of larger piston)^2
Step 5: Set up an equation using the fact that the pressure on both pistons is the same.
Pressure on the larger piston = Pressure on the smaller piston
Pressure on the larger piston = Force / Area of larger piston
79577.47 Pa = 100 N / (π * (radius of larger piston)^2)
Step 6: Solve the equation for the radius of the larger piston.
(radius of larger piston)^2 = 100 N / ( π * 79577.47 Pa)
radius of larger piston ≈ √(100 N / ( π * 79577.47 Pa))
Now, let's plug in the values and calculate it:
radius of larger piston ≈ √(100 N / ( π * 79577.47 Pa))
radius of larger piston ≈ √(0.001257 m^2)
radius of larger piston ≈ 0.0355 m
Therefore, the radius of the larger piston should be approximately 0.0355 meters, or 3.55 cm, in order to lift a 1,700 kg car with a moderate force of 100 N.