The hydraulic oil in a car lift has a density of 8.54 x 102 kg/m3. The weight of the input piston is negligible. The radii of the input piston and output plunger are 5.01 x 10-3 m and 0.160 m, respectively. What input force F is needed to support the 21900-N combined weight of a car and the output plunger, when (a) the bottom surfaces of the piston and plunger are at the same level, and (b) the bottom surface of the output plunger is 1.10 m above that of the input plunger?
To find the input force needed to support the weight of the car and the output plunger, we need to consider two scenarios:
(a) When the bottom surfaces of the piston and plunger are at the same level.
(b) When the bottom surface of the output plunger is 1.10 m above that of the input plunger.
Let's start by calculating the pressure exerted by the hydraulic oil:
(a) When the bottom surfaces of the piston and plunger are at the same level:
1. The area of the input piston can be calculated using the formula: A = πr^2, where r is the radius of the input piston.
Substituting the given value, A = π (5.01 x 10^-3 m)^2.
2. The pressure exerted by the hydraulic oil is given by the formula: P = ρgh, where ρ is the density of the oil, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height difference between the two surfaces.
In this case, the height difference is zero, so h = 0.
3. The weight of the car and the output plunger is equal to the force exerted, F = 21900 N.
Now we can calculate the input force F:
F = P × A
= ρgh × A
Substituting the given values, the formula becomes:
F = (8.54 x 10^2 kg/m^3) × (9.8 m/s^2) × (π (5.01 x 10^-3 m)^2)
Simplifying this expression will give us the input force F when the bottom surfaces of the piston and plunger are at the same level.
(b) When the bottom surface of the output plunger is 1.10 m above that of the input plunger:
In this scenario, the height difference, h, is 1.10 m. We can follow the same steps as above to calculate the input force F.
F = ρgh × A
Substituting the given values:
F = (8.54 x 10^2 kg/m^3) × (9.8 m/s^2) × (π (5.01 x 10^-3 m)^2)
Remember to multiply the final result by the weight of the car and the output plunger (21900 N) to get the final answer.
I hope this explanation helps you understand how to calculate the input force required in both scenarios of the car lift.