A 12,000-N car is raised using a hydraulic lift,which consists of a U-tube with arms of unequal areas,filled with incompressible oil with a density of 800 kg/m³ and capped at both ends with tight-fitting pistons.The wider arm of the U-tube has a radius of 18.0 cm and the narrower arm has a radius of 5.00 cm.The car rests on the piston on the wider arm of the U-tube.The pistons are initially at the same level.What is the force that must be applied to the smaller piston in order to lift the car after it has been raised 1.20 m? (For purposes of this problem,you can neglect the weight of the pistons.) A) 0.954 kN B) 1.88 kN C) 1.96 kN D) 3.67 kN E) 1.20 kN

Question

Quizplus · Accepted Answer

First, we need to find the pressure in the system. Since the pistons are at the same level initially, the pressure at the bottom of both sides of the U-tube is atmospheric pressure, 1.013x10^5 N/m^2. Since the oil is incompressible, the pressure at any point in the oil must be the same on both sides of the U-tube. Using Pascal's Law, this means that the pressure at the piston on the larger arm of the U-tube is the same as the pressure at the bottom of the smaller arm.

The pressure at the bottom of the smaller arm can be found using the equation:

P = ρgh

where P is pressure, ρ is density, g is acceleration due to gravity, and h is the height of the oil column above the piston in the smaller arm.

The height of the oil column can be found using the fact that the volume of oil in the U-tube is constant. This means that the product of the area and height of the oil column on each side of the U-tube is the same. Letting h1 and h2 be the heights of the oil column in the larger and smaller arms respectively, we have:

A1h1 = A2h2

where A1 is the area of the larger piston and A2 is the area of the smaller piston. Rearranging this equation, we get:

h2 = (A1/A2)h1

Plugging in the values given, we get:

h2 = (π(0.18^2)/(π(0.05^2))) x 1.20 m = 25.92 m

Now we can use the equation for pressure to find the pressure at the bottom of the smaller arm:

P = ρgh = (800 kg/m^3)(9.81 m/s^2)(25.92 m) = 2.01x10^5 N/m^2

This is the pressure that must be exerted on the oil in the smaller arm in order to lift the car. We can find the force needed to create this pressure using the equation:

F = PA

where F is the force needed, P is the pressure, and A is the area of the smaller piston. Plugging in the values given, we get:

F = (2.01x10^5 N/m^2)(π(0.05^2)) = 1.96 kN

Therefore, the best choice is C.

A 12,000-N Car Is Raised Using a Hydraulic Lift,which Consists