A hydraulic lift has two connected pistons with cross-sectional areas 5 cm2 and 350 cm2. It is filled with oil of density 740 kg/m3.
a) What mass must be placed on the small piston to support a car of mass 1400 kg at equal fluid levels?
20 OK
HELP: The pressure is constant at any height.
b) With the lift in balance with equal fluid levels, a person of mass 100 kg gets into the car. What is the equilibrium height difference in the fluid levels in the pistons?
HELP: Balance the pressure from the weight of the fluid and the pressure from the person's weight.
c) How much did the height of the car drop when the person got in the car?
HELP: The fluid is incompressible, so volume is conserved.
To solve this problem, we can use the principles of Pascal's law and Archimedes' principle.
a) To determine the mass that must be placed on the small piston to support the car, we need to find the pressure exerted on the fluid by the car's weight.
The pressure exerted on a fluid is given by the formula P = ρgh, where P is the pressure, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the height of the fluid column.
Since the fluid levels are equal, the height h is the same for both pistons. Let's calculate the pressure exerted on the fluid by the car's weight (in N/m^2 or Pascal):
P_car = ρ * g * h = 740 kg/m^3 * 9.8 m/s^2 * h
Now, we can use Pascal's law, which states that the pressure exerted at any point in an incompressible fluid is transmitted equally in all directions. Thus, the pressure on the small piston must be equal to the pressure on the large piston for the system to be in equilibrium.
The pressure exerted on the small piston is given by the formula P_small = F_small / A_small, where F_small is the force exerted on the small piston and A_small is its cross-sectional area.
The pressure exerted on the large piston is given by the formula P_large = F_large / A_large, where F_large is the force exerted on the large piston and A_large is its cross-sectional area.
Since the system is in equilibrium, we have P_small = P_large. Let's calculate the force exerted on the large piston:
F_large = P_small * A_large = P_car * A_small * (A_large / A_small)
Now, to find the mass that must be placed on the small piston to support the car, we need to multiply the force exerted on the large piston by the acceleration due to gravity:
mass = F_large / g
Substituting the values we have:
mass = (P_car * A_small * (A_large / A_small)) / g
Substituting the given values:
mass = (740 kg/m^3 * 9.8 m/s^2 * h * 5 cm^2 * (350 cm^2 / 5 cm^2)) / 9.8 m/s^2
mass = h * 740 kg/m^3 * 350 cm^2
So, the mass that must be placed on the small piston to support the car is h * 740 kg/m^3 * 350 cm^2.
b) To find the equilibrium height difference in the fluid levels when the person gets into the car, we need to consider the additional weight added to the system.
First, let's calculate the weight of the person:
weight_person = mass_person * g
Now, let's calculate the additional pressure exerted on the fluid by the person's weight:
P_person = weight_person / A_small = (mass_person * g) / A_small
Since the system is in equilibrium, the pressure on the small piston must be equal to the pressure on the large piston, considering both the car's weight and the person's weight.
P_small = P_large + P_person
Let's solve for the equilibrium height difference h_person:
h_person * 740 kg/m^3 * 5 cm^2 = (740 kg/m^3 * 9.8 m/s^2 * h * 350 cm^2) + (mass_person * g)
Now, to find the equilibrium height difference, we can solve for h_person:
h_person = [(740 kg/m^3 * 9.8 m/s^2 * h * 350 cm^2) + (mass_person * g)] / (740 kg/m^3 * 5 cm^2)
Substituting the given values:
h_person = [(740 kg/m^3 * 9.8 m/s^2 * h * 350 cm^2) + (100 kg * 9.8 m/s^2)] / (740 kg/m^3 * 5 cm^2)
So, the equilibrium height difference in the fluid levels is [(740 kg/m^3 * 9.8 m/s^2 * h * 350 cm^2) + (100 kg * 9.8 m/s^2)] / (740 kg/m^3 * 5 cm^2).
c) To find how much the height of the car dropped when the person got into the car, we can subtract the equilibrium height difference h_person from the initial height h.
height_drop = h - h_person
Substituting the given values and the expression for h_person obtained in part b:
height_drop = h - [(740 kg/m^3 * 9.8 m/s^2 * h * 350 cm^2) + (100 kg * 9.8 m/s^2)] / (740 kg/m^3 * 5 cm^2)
To solve this problem, we will use the principles of Pascal's law, which states that the pressure exerted at any point on a confined fluid is transmitted undiminished to all other points in the fluid and to the walls of the container.
a) To determine the mass that must be placed on the small piston to support the car, we need to find the pressure exerted by the car and equate it to the pressure on the small piston.
1. Start by calculating the pressure exerted by the car. We know that the density of the oil is 740 kg/m3, and the mass of the car is 1400 kg. The pressure is given by the formula:
Pressure = Density * Acceleration due to gravity * Height
Since the car is at the same height as the fluid levels, the height term cancels out, and we can simply use:
Pressure = Density * Acceleration due to gravity
2. Now we need to find the pressure on the small piston. The pressure is given by the formula:
Pressure = Force / Area
Since we want to find the mass that needs to be placed on the small piston, we can rewrite the formula as:
Force = Mass * Acceleration due to gravity
3. Equate the two pressures:
Pressure exerted by the car = Pressure on the small piston
(Density * Acceleration due to gravity) = (Mass * Acceleration due to gravity) / Area
Cancel out the acceleration due to gravity:
Density = Mass / Area
Rearranging the equation:
Mass = Density * Area
Plug in the values: Density = 740 kg/m3, Area = 5 cm2 (convert to m2)
Mass = 740 kg/m3 * 5 cm2 * (1 m2/10,000 cm2) = 0.37 kg
Therefore, a mass of 0.37 kg must be placed on the small piston to support the car of mass 1400 kg at equal fluid levels.
b) To find the equilibrium height difference in the fluid levels when a person of mass 100 kg gets into the car, we need to balance the pressures exerted by the weight of the fluid and the person's weight.
1. Calculate the pressure exerted by the person's weight:
Pressure = Density * Acceleration due to gravity
2. Calculate the pressure on the large piston using the formula:
Pressure = Force / Area
3. Equate the two pressures:
Pressure exerted by the person's weight = Pressure on the large piston
(Density * Acceleration due to gravity) = (Mass * Acceleration due to gravity) / Area
Cancel out the acceleration due to gravity:
Density = Mass / Area
Rearranging the equation:
Mass = Density * Area
Plug in the values: Density = 740 kg/m3, Area = 350 cm2 (convert to m2)
Mass = 740 kg/m3 * 350 cm2 * (1 m2/10,000 cm2) = 25.9 kg
Therefore, the mass of the fluid is 25.9 kg.
c) Since the fluid is incompressible, the volume is conserved. Therefore, the volume of the fluid displaced by the person entering the car is equal to the volume of the car that dropped in height.
1. Find the volume of the person:
Volume = Mass / Density = 100 kg / 740 kg/m3 = 0.135 m3
2. The volume of the fluid displaced is equal to the volume of the person:
Volume of fluid displaced = 0.135 m3
3. Use the formula for volume of a cylinder:
Volume = Base area * Height
Height = Volume / Base area = 0.135 m3 / 350 cm2 * (1 m2/10,000 cm2) = 0.00000039 m
Therefore, the height of the car dropped by approximately 0.00000039 m when the person got into the car.