In a hydraulic lift, the radii of the pistons are 2.93 cm and 10.1 cm. A car weighing W = 11.0 kN is to be lifted by the force of the large piston.

(a) What force Fa must be applied to the small piston? N

(b) When the small piston is pushed in by 10.4 cm, how far is the car lifted?
mm

(c) Find the mechanical advantage of the lift, which is the ratio W / Fa.

I really don't understand how to do this please explain step for step how to do this! I would really appreciate it, thank you.

a.

F1/A1 =F2/A2,
A =πR^2,
Fa = W•(r/R)^2 = 11000•(2.93/10.1)^2 = =925.7 N
b.
h2 = h1•(A1/A2) = h1•(r/R) = 3.02 cm
b.
MA = W/Fa = 11000/925.7 =11.9

To solve this problem step-by-step, we can use Pascal's law for the hydraulic lift. Pascal's law states that the pressure in a fluid at rest is transmitted equally in all directions. This means that the pressure exerted on the small piston is equal to the pressure exerted on the large piston. We can use this principle to solve the problem.

Step 1: Calculate the force exerted on the large piston.
Given:
Radius of the large piston (R1) = 10.1 cm = 0.101 m (since 1 cm = 0.01 m)
Weight of the car (W) = 11.0 kN = 11,000 N (since 1 kN = 1000 N)

Formula:
Force = Pressure x Area

Since the pressure is the same on both pistons, we only need to calculate the force exerted on the large piston.

Area of the large piston (A1) = π * (R1^2)
= π * (0.101^2)

Force on the large piston (F1) = Pressure x Area
F1 = P x A1
We need to find F1.

Step 2: Calculate the force exerted on the small piston.
Given:
Radius of the small piston (R2) = 2.93 cm = 0.0293 m

Similarly, we can calculate the area of the small piston.

Area of the small piston (A2) = π * (R2^2)
= π * (0.0293^2)

Step 3: Apply Pascal's law.
Pascal's law states that the ratio of the forces exerted on the pistons is equal to the ratio of the areas of the pistons.

Let Fa be the force applied to the small piston (F2).

Fa / F1 = A2 / A1

Step 4: Solve for Fa.
Rearrange the equation to solve for Fa:

Fa = (F1 * A2) / A1

Substitute the values into the equation:

Fa = (F1 * A2) / A1
Fa = (F1 * π * (0.0293^2)) / (π * (0.101^2))

Simplify the equation:

Fa = (F1 * 0.00085249) / 0.010201

Step 5: Calculate Fa.
Substitute the value of F1 into the equation:

Fa = (11,000 * 0.00085249) / 0.010201

Solve:
Fa ≈ 915.92 N

Therefore, the force Fa applied to the small piston is approximately 915.92 N.

Now, let's move on to part (b) and (c) of the problem.

Step 6: Calculate the displacement of the car.
Given:
Displacement of the small piston (d2) = 10.4 cm = 0.104 m

Using the principle of similar triangles, we can calculate the displacement (d1) of the large piston.

d2 / d1 = A2 / A1

Solve for d1:

d1 = (d2 * A1) / A2

Substitute the values into the equation:

d1 = (0.104 * π * (0.101^2)) / (π * (0.0293^2))

Simplify the equation:

d1 = (0.104 * 0.010201) / 0.00085249

Step 7: Calculate d1.
Solve the equation:

d1 ≈ 1.241 m

Therefore, when the small piston is pushed in by 10.4 cm, the car is lifted approximately 1.241 m.

Step 8: Calculate the mechanical advantage.
The mechanical advantage is the ratio of the weight of the car (W) to the force applied to the small piston (Fa).

Mechanical Advantage (MA) = W / Fa

Substitute the values into the equation:

MA = 11,000 / 915.92

Solve the equation:

MA ≈ 12.00

Therefore, the mechanical advantage of the lift is approximately 12.00.

Sure! I can help you understand how to solve this problem step by step.

(a) To find the force Fa that must be applied to the small piston, we can use Pascal's principle, which states that the pressure applied to an enclosed fluid is transmitted equally in all directions.

The formula to calculate force using pressure is: Force = Pressure * Area.

First, let's calculate the pressure on the large piston, which is the same as the pressure on the small piston since they are in a closed hydraulic system.
Given that the radius of the small piston is 2.93 cm, and the radius of the large piston is 10.1 cm, we can calculate the pressure ratio as follows:

Pressure ratio = (Area of large piston) / (Area of small piston)
= (π * (10.1 cm)^2) / (π * (2.93 cm)^2)
= (10.1 cm)^2 / (2.93 cm)^2

Now, we know that pressure is force divided by area:
Pressure ratio = (Force applied to large piston) / (Force applied to small piston)
Force applied to large piston = W (weight of the car) = 11.0 kN

Hence, we can rearrange the equation to find the force Fa applied to the small piston:
Fa = (Force applied to large piston) / Pressure ratio

Now, substitute the given values to calculate Fa:
Fa = 11.0 kN / (10.1 cm)^2 / (2.93 cm)^2
Fa = 11.0 kN / (10.1)^2 / (2.93)^2

(b) To find how far the car is lifted when the small piston is pushed in by 10.4 cm, we can use the ratios of the radii of the two pistons.

The ratio of the distances moved by the pistons is the inverse ratio of their radii:
Distance ratio = (Radius of small piston) / (Radius of large piston)
= 2.93 cm / 10.1 cm

Now, we can calculate how far the car is lifted:
Distance lifted = Distance ratio * Distance moved by small piston
= (2.93 cm / 10.1 cm) * 10.4 cm

(c) The mechanical advantage of the lift is the ratio of the weight of the car (W) to the force applied to the small piston (Fa):
Mechanical advantage = W / Fa

Now, substitute the given values to calculate the mechanical advantage:
Mechanical advantage = 11.0 kN / Fa

I hope this step-by-step explanation helps you understand how to approach this problem. Let me know if you have any further questions!

Thanks for your help, I really appreciate it! The only thing I was wondering about is part b) it says its wrong i know it's 3.02 cm, but the answer has to be in mm i got that that would be 30.2 mm, but still its wrong can you please say why I am getting it wrong?