In: Physics
The hydraulic oil in a car lift has a density of 8.30 x 102 kg/m3. The weight of the input piston is negligible. The radii of the input piston and output plunger are 7.30 x 10-3 m and 0.136 m, respectively. What input force F is needed to support the 23800-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 1.00 m above that of the input plunger?
a) When the piston and the plunger are at the same level,
according to the Pascal's principle
F1 / A1 = F2 / A2
Where F is the force and A is the surface area .The subscripts 1
and 2 represents the piston and the plunger respectively
The combined force on the output plunger due to the weight
F2 = 23800 N
Area of the output plunger A2 =
r2 = 3.14 x (0.136 m)2
A2 = 0.058 m2
The area of the input piston A1 =
r2 = 3.14 x (7.30 x 10-3
m)2
A1 = 0.167 x 10-3
m2
F1 = (F2 / A2 ) x
A1
F1 = (23800 / 0.058) x 0.167 x 10-3
F1 = 68.53 N
This is the input force needed
b) We know that the pressure at a height h is
P2 = P1+ g h
Where P2is the pressure at the plunger ,P1is
the pressure at the piston and h is the height, is the density
of the lliqid
F2/ A2 = ( F1 /
A1) + gh
We need to find F1
F1 = ( (F2/ A2) - gh )
A1
F1 = ((23800 / 0.058) - (8.30 x 102 x 9.81 x
1.00)) x 0.167 x 10-3
F1 = 67.17 N