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?
To find the radius of the larger piston, we can use the principle of Pascal's law, which states that the pressure applied to an enclosed fluid is transmitted equally in all directions. This law allows us to calculate the force exerted by the larger piston based on the force applied to the smaller piston.
Here's how we can calculate the radius of the larger piston:
Step 1: Calculate the area of the smaller piston.
- Recall that the area of a circle is given by the formula A = π * r^2, where A is the area and r is the radius.
- Given that the radius of the smaller piston is 2 cm, we can calculate its area as A_smaller = π * (0.02 m)^2.
Step 2: Calculate the pressure generated on the hydraulic fluid by the smaller piston.
- Using Pascal's law, the pressure applied on the smaller piston will be transmitted equally to the larger piston.
- Since the force applied to the smaller piston is 100 N and the area is A_smaller, the pressure on the fluid can be calculated as P = 100 N / A_smaller.
Step 3: Calculate the area of the larger piston.
- Since the pressure is transmitted equally in all directions, the pressure on the larger piston is equal to P.
- Also, the force exerted by the larger piston can be calculated as F_larger = P * A_larger, where A_larger is the area of the larger piston.
- We are given that the mass of the piston and platform is 300 kg, so the weight of the piston and platform is W = m * g, where g is the acceleration due to gravity (approximated as 9.8 m/s^2).
- Since the hydraulic lift is in equilibrium, the force exerted by the larger piston, F_larger, is equal to the weight, W.
- Therefore, P * A_larger = W.
- Solving for A_larger, we get A_larger = W / P.
Step 4: Calculate the radius of the larger piston.
- Now that we have the area of the larger piston, A_larger, we can calculate its radius by rearranging the formula for the area of a circle:
A_larger = π * r_larger^2.
- Rearranging the equation, we get r_larger = √(A_larger / π).
Using the above steps, you can calculate the radius of the larger piston.