An automobile having a mass of 1750 kg is placed on a hydraulic lift in a garage. The piston lifting the car is 0.244 m in diameter. A mechanic attaches a pumping mechanism to a much smaller piston, 1.50 cm in diameter, which is connected by hydraulic lines to the lift. She pumps the handle up and down, slowly lifting the car. What is the force exerted on the small piston during each downward stroke?
To find the force exerted on the small piston during each downward stroke, we can use Pascal's law which states that the pressure exerted on a fluid is transmitted uniformly in all directions. We can start by finding the pressure exerted on the large piston due to the weight of the car.
Step 1: Find the area of the large piston.
The diameter of the large piston is given as 0.244 m. We can calculate the area using the formula:
Area = π * (radius)^2
Radius = (diameter) / 2
= 0.244 m / 2
= 0.122 m
Area = π * (0.122 m)^2
Step 2: Find the pressure on the large piston.
The weight of the car exerted on the large piston creates a force which in turn creates pressure on the piston. We can calculate the pressure using the formula:
Pressure = Force / Area
First, we need to find the weight of the car using the formula:
Weight = mass * acceleration due to gravity
The mass of the car is given as 1750 kg, and acceleration due to gravity is approximately 9.8 m/s^2.
Weight = 1750 kg * 9.8 m/s^2
Then, we can find the force exerted on the large piston using the formula:
Force = Weight
Note that the force is equal to the weight.
Pressure = Force / Area
Step 3: Find the force exerted on the small piston.
Now that we have found the pressure on the large piston, we can use Pascal's law to find the force exerted on the small piston.
According to Pascal's law, the pressure is transmitted uniformly in all directions.
Pressure on the small piston = Pressure on the large piston
Now, let's find the area of the small piston.
The diameter of the small piston is given as 1.50 cm. We can calculate the area using the formula:
Area = π * (radius)^2
First, convert 1.50 cm to meters:
1 cm = 0.01 m
Radius = (diameter) / 2
= 0.015 m / 2
= 0.0075 m
Area = π * (0.0075 m)^2
Finally, we can calculate the force exerted on the small piston using the formula:
Force = Pressure * Area
Substitute the values into the formula to find the force exerted on the small piston during each downward stroke.
To calculate the force exerted on the small piston during each downward stroke, we can use Pascal's law which states that the pressure exerted in a fluid is transmitted equally in all directions.
First, let's convert the diameter of the small piston to meters:
Diameter of small piston = 1.50 cm = 0.015 m
Next, we can calculate the area of the small piston:
Area of small piston = (π/4) x (diameter of small piston)^2
= (π/4) x (0.015 m)^2
Now, we need to determine the force exerted on the small piston during each downward stroke. The force can be calculated using the formula:
Force = Pressure x Area
However, we need to find the pressure exerted on the fluid by the pumping mechanism. Since the pressure is transmitted equally in all directions, it will be the same as the pressure in the hydraulic lift.
The pressure in the hydraulic lift can be calculated using the formula:
Pressure = Force / Area
To find the pressure in the hydraulic lift, we need to consider the dimensions of the large piston as well. Unfortunately, the diameter of the large piston is not mentioned in the question. Without that information, we cannot accurately determine the force exerted on the small piston during each downward stroke.
Please note that in order to provide a specific answer, we require the diameter of the large piston or additional information to compute the force exerted on the small piston during each downward stroke.