The two pistons in a hydraulic lift have radii of 2.67 cm and 20.00 cm respectively. What force must be applied to the 2.67 cm piston so that a 2.0 x 103 kg mass resting on the 20.0 cm piston is lifted?

To calculate the force required to lift a mass in a hydraulic lift, we can use Pascal's law, which states that the pressure exerted in an enclosed fluid is transmitted equally in all directions.

Here's how we can solve the problem step by step:

Step 1: Calculate the pressure exerted on the 20.0 cm piston due to the weight of the mass.

The pressure (P) exerted by a fluid is given by the formula:

P = F/A

where F is the force and A is the area.

The area of the 20.0 cm piston can be calculated using the formula:

A = π * r^2

where r is the radius of the piston.

Substituting the values:

r = 20.00 cm = 0.20 m
A = π * (0.20)^2 = 0.1257 m^2

Next, we need to calculate the force (F) exerted on the 20.0 cm piston due to the mass.

F = m * g

where m is the mass and g is the acceleration due to gravity.

Substituting the values:

m = 2.0 x 10^3 kg
g = 9.8 m/s^2

F = (2.0 x 10^3 kg) * (9.8 m/s^2) = 1.96 x 10^4 N

Therefore, the force exerted on the 20.0 cm piston due to the weight of the mass is 1.96 x 10^4 N.

Step 2: Calculate the force required to lift the mass using the smaller piston.

According to Pascal's law, the pressure is the same at all points in an enclosed fluid. Therefore, the pressure exerted by the small piston is the same as the pressure exerted by the large piston:

P_small = P_large

Since P = F/A, we can rewrite this as:

F_small / A_small = F_large / A_large

Substituting the values:

F_small / (π * (0.0267 m)^2) = (1.96 x 10^4 N) / (0.1257 m^2)

Solving for F_small, we get:

F_small = (π * (0.0267 m)^2) * (1.96 x 10^4 N) / (0.1257 m^2)

F_small ≈ 118 N

Therefore, the force that must be applied to the 2.67 cm piston to lift the 2.0 x 10^3 kg mass is approximately 118 N.