In a hudraulic lift, the radii of the pistons are 2.50 cm and 10.0 cm. A car weighing 10.0 kN is to be lifted by the force of the large piston. (a) what force must be applied to the small piston? (b) when the small piston is pushed in by 10.0 cm, how far is the car lifted? (c) Find the mechanical advantage of the lift, which is the ratio.

To solve this problem, we need to apply Pascal's Law, which states that the pressure applied to a fluid is transmitted uniformly in all directions. Here's how you can find the answers to each part of the question:

(a) To determine the force applied to the small piston, we can use the principle of Pascal's Law. According to Pascal's Law, the pressure applied at one point in an incompressible fluid is distributed equally throughout the fluid. This can be expressed by the equation:

Pressure (P) = Force (F) / Area (A)

First, we need to calculate the area of each piston. The area of a circle is given by the formula A = πr^2, where r is the radius of the circle.

For the large piston:
Radius (r1) = 10.0 cm = 0.1 m
Area (A1) = π(0.1 m)^2

For the small piston:
Radius (r2) = 2.50 cm = 0.025 m
Area (A2) = π(0.025 m)^2

Now, since the pressure is the same in both pistons, we can set up an equation:

P1 = P2
F1 / A1 = F2 / A2

We know that the force applied to the large piston (F1) is 10.0 kN (given), so we can substitute the values:

10,000 N / A1 = F2 / A2

Rearranging the equation, we can solve for F2:

F2 = (10,000 N / A1) * A2

Substitute the values and calculate F2:

F2 = (10,000 N / (π(0.1 m)^2)) * (π(0.025 m)^2)

(b) To find how far the car is lifted (d), we can use the principle of Pascal's Law again. This time, we need to relate the change in height (d) of the small piston to the change in height (D) of the car.

Since the cross-sectional area of the small piston is smaller than that of the large piston, each unit of volume displaced by the small piston corresponds to a larger displacement in height for the car.

Using the equation:

A1 * D1 = A2 * D2

where A1 and A2 are the cross-sectional areas of the large and small pistons, and D1 and D2 are the corresponding changes in height, we can solve for D2:

D2 = (A1 * D1) / A2

We're given that the small piston is pushed in by 10.0 cm (D1 = -0.10 m), so we can substitute the values and calculate D2:

D2 = (A1 * -0.10 m) / A2

(c) The mechanical advantage of the lift is the ratio of the force applied to the large piston (F1) to the force applied to the small piston (F2). Therefore, we can calculate it by dividing F1 by F2:

Mechanical Advantage = F1 / F2

Substitute the values calculated in part (a) to find the mechanical advantage.