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 10^3 kg mass resting on the 20.0 cm piston is lifted?
To find out the force required to lift the mass using the hydraulic lift, we need to utilize Pascal's law, which states that pressure in a fluid is transmitted equally in all directions.
We can set up an equation using the principles of pressure:
Pressure1 = Pressure2
Pressure = Force / Area
Since the force is unknown for the smaller piston and known for the larger piston, we'll set up the equation as follows:
Force1 / Area1 = Force2 / Area2
The area is calculated using the formula for the surface area of a circle:
Area = π * radius^2
Let's calculate the areas for the two pistons:
Area1 = π * (2.67 cm)^2
Area2 = π * (20.00 cm)^2
Now, we can plug in the given values:
Area1 = 3.14 * (2.67 cm)^2
≈ 22.33 cm^2
Area2 = 3.14 * (20.00 cm)^2
≈ 1256.00 cm^2
Now, we can rearrange the equation to solve for Force1 (the force required to lift the mass):
Force1 = (Area1 / Area2) * Force2
Substituting the given values:
Force1 = (22.33 cm^2 / 1256.00 cm^2) * (2.0 x 10^3 kg * 9.8 m/s^2)
≈ 35.48 kg * 9.8 m/s^2
≈ 348.30 N
Therefore, the force required to lift the mass using the 2.67 cm piston is approximately 348.30 Newtons.