A car of mass 1390 kg rests on a hydraulic lift with a piston of radius 0.26 m that is connected to a second piston of radius 0.033 m. If the car is just barely lifted off the ground, what is the force at the second piston?
I Tried
F1/F2 = r1^2 / r2^2
==> 1390x10 / F2 = 0.26^2 / 0.033^2 then I calculated F2 and found 223.92N but it didn't work.. Help please
Your approach is correct, but it seems you made a mistake in the calculation. Let's go through the steps together:
1. Use the formula for the hydraulic lift: F1/F2 = (r2^2)/(r1^2)
- F1 is the force applied to the first piston (known as the effort force).
- F2 is the force at the second piston (known as the load force).
- r1 is the radius of the first piston.
- r2 is the radius of the second piston.
2. Plug in the given values:
- F1 is the weight of the car, which is mass x gravity. Given that the mass of the car is 1390 kg and gravity is approximately 10 m/s^2, we have F1 = 1390 kg x 10 m/s^2 = 13,900 N.
- r1 = 0.26 m (radius of the larger piston).
- r2 = 0.033 m (radius of the smaller piston).
3. Substitute these values into the formula: F1/F2 = (r2^2)/(r1^2)
- 13,900 N / F2 = (0.033 m)^2 / (0.26 m)^2
4. Now, cross multiply and solve for F2:
- F2 = (13,900 N) / ((0.033 m)^2 / (0.26 m)^2)
- F2 = (13,900 N) / (0.001089 m^2 / 0.0676 m^2)
- F2 = (13,900 N) / (0.016077)
- F2 ≈ 865.51 N
Therefore, the force at the second piston is approximately 865.51 N, not 223.92 N as you initially calculated.
To calculate the force at the second piston, you can use Pascal's law, which states that the pressure exerted at any point in an enclosed fluid is transmitted equally in all directions.
First, let's calculate the pressure at the first piston. The force at the first piston is given by:
F1 = m * g
where m is the mass of the car (1390 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2):
F1 = 1390 kg * 9.8 m/s^2
F1 = 13622 N
The area of the first piston is:
A1 = π * r1^2
where r1 is the radius of the first piston (0.26 m):
A1 = π * (0.26 m)^2
A1 ≈ 0.212 m^2
Now, we can calculate the pressure at the first piston:
P1 = F1 / A1
P1 = 13622 N / 0.212 m^2
P1 ≈ 64163 Pa
According to Pascal's law, this pressure is transmitted to the second piston, so the force at the second piston can be determined using the area of the second piston:
A2 = π * r2^2
where r2 is the radius of the second piston (0.033 m):
A2 = π * (0.033 m)^2
A2 ≈ 0.0034 m^2
Now, we can calculate the force at the second piston:
F2 = P1 * A2
F2 ≈ 64163 Pa * 0.0034 m^2
F2 ≈ 218.61 N
Therefore, the force at the second piston is approximately 218.61 N.