Question

A hydraulic lift has two connected pistons with cross-sectional areas 10 cm² and 490 cm². It is filled with oil of density 700 kg/m³.

1. What mass must be placed on the small piston to support a car of mass 10³ kg at equal fluid levels?

m = 20.41kg

2. With the lift in balance with equal fluid levels, a person of mass 60 kg gets into the car. What is the equilibrium height difference in the fluid levels in the pistons?

?h =

3. How much did the height of the car drop when the person got in the car?

h =

Answer 1

Area on Small piston = As =10
Area on large piston = Al = 490

Pressure = Force / Area.
To maintain equal fluid levels pressure has to be equal on both piston.

a)

Force on Large piston due to 1000 kg = 1000*9.81 = 9810 N
Pressure on large piston = Force/Area = 9810/490 = 20.02 N/cm2

Pressure on small piston = Pressure on large piston = 20.02 N/cm2
Weight on piston = Ws

20.02 = Ws/10
Ws = 200.2 N

Mass to be placed on small piston = 200.2/9.81 = 20.40 kg

b)

We know the car is placed on the Large piston. Now the person enters the car.

Weight of the person in the car will apply an extra pressure on the large piston.

Pressure due to Person = Weight of person/area of large piston
= 60*9.81/490
= 1.20122 N/cm2

Now this pressure on the Large piston side will push the oil upwards to a certain height on the small piston side. In other words to counter this weight due to person on the Large piston side, an equal force has to be applied by the weight of the oil in the small piston column.

Pressure due to person = Pressure due to oil raise in the small piston column

Lets find pressure due to oil raise in the small piston column.
Pressure = force/area

force = weight = Mass*9.81
Mass = Volume *density

Volume = Area * h
Area = 10 cm2
h = ? in cm
density = 700 kg/m3 = 700* 10^-6 kg/cm3

Force due to weight of oil in column = 10 *h *700* (10^6)*9.81
Pressure due to oil = 10 *h * 700* (10^-6)*9.81/ 10
= [6.867 *(10^-3) * h] N/cm2

This value should be equal to the pressure due to person = 1.20122
N/cm2

1.20122 = 6.867*10^-3 *h
h = 174.93 cm


c) Now to determine the height of the car drop.

Volume of oil displaced from large piston = Volume of oil displaced into small piston.

Area of Large piston * height of large piston = Area of small piston * height of small piston
Al * hl = As * hs
490 * hl = 10 * 174.92
hl = 3.57 cm