If the input piston of a hydraulic jack has a radius of 0.005 m, what force is necessary to apply to it to lift a 1500-kg car on the output piston whose radius is 0.5 m?
Nine different identities. You poor soul.
p= F₁/A₁= F₂/A₂
F₁/π R₁²= F₂/π R₂²
F₁ = F₂ R₁²/R₂²= m₂g •R₁²/R₂²=
=1500•9.8 •0.005²/0.5²=1.47 N
To calculate the force necessary to lift the car using a hydraulic jack, you can use Pascal's law, which states that the pressure exerted on an enclosed fluid will be transmitted equally in all directions.
Step 1: Calculate the pressure on the input piston.
The formula for pressure is P = F/A, where P is pressure, F is the force exerted, and A is the area.
Given:
- Radius of the input piston (r1) = 0.005 m
To calculate the area of the input piston (A1), use the formula for the area of a circle: A = πr^2.
A1 = π * (r1)^2
Step 2: Calculate the area on the output piston.
Given:
- Radius of the output piston (r2) = 0.5 m
To calculate the area of the output piston (A2), use the same formula as in Step 1.
A2 = π * (r2)^2
Step 3: Calculate the force needed to lift the car.
Given:
- Mass of the car (m) = 1500 kg
Using Pascal's law, the pressure on both the input and output pistons will be the same. Therefore, the force exerted on the input piston (F1) will be equal to the force exerted on the output piston (F2).
Using the formula for pressure from Step 1:
P = F/A
The pressure on the input piston (P1) is equal to the pressure on the output piston (P2):
P1 = P2
Substituting the values from Steps 1 and 2 into the formula:
F1/A1 = F2/A2
Rearranging the equation to solve for F2:
F2 = (F1 * A2) / A1
Substituting the given values:
F2 = (F1 * π * (r2)^2) / (π * (r1)^2)
Step 4: Calculate the force required.
To lift the car, we need to find F2, which is the force exerted on the output piston.
Substituting the given values and solving:
F2 = (F1 * π * (0.5)^2) / (π * (0.005)^2)
F2 = (F1 * 0.25) / 0.000025
Simplifying:
F2 = 10,000 * F1
Since F1 is the force exerted on the input piston, we can solve for it using the equation F = m * g, where F is the force, m is the mass, and g is the acceleration due to gravity. Assuming g = 9.8 m/s^2, we can calculate F1.
Given:
- m = 1500 kg
- g = 9.8 m/s^2
F1 = m * g
F1 = 1500 kg * 9.8 m/s^2
Simplifying:
F1 = 14,700 N
Finally, we can calculate the force required to lift the car by substituting the value of F1 into the equation for F2:
F2 = 10,000 * F1
F2 = 10,000 * 14,700 N
Calculating:
F2 = 147,000,000 N
Therefore, the force necessary to apply to lift the 1500-kg car on the output piston is 147,000,000 Newtons.
To find the force necessary to lift the car using the hydraulic jack, we can apply Pascal's law, which states that pressure exerted in an enclosed fluid is transmitted equally in all directions.
Step 1: Find the areas of both pistons.
The area of the input piston (A1) can be calculated using the formula for the area of a circle: A1 = π * r1^2, where r1 is the radius of the input piston.
Here, r1 = 0.005 m, so A1 = π * (0.005)^2.
The area of the output piston (A2) can be calculated in the same way: A2 = π * r2^2, where r2 is the radius of the output piston.
In this case, r2 = 0.5 m, so A2 = π * (0.5)^2.
Step 2: Find the pressure (P) exerted on the input and output pistons.
According to Pascal's law, the pressure exerted on both pistons is equal. We can express this as P1 = P2 = P.
Step 3: Calculate the force (F1) applied on the input piston.
Force is defined as the product of pressure and area: F = P * A.
Here, P = P1, and A = A1.
So, F1 = P1 * A1.
Step 4: Calculate the force (F2) exerted on the output piston.
Force is again defined as the product of pressure and area: F = P * A.
Now, P = P2, and A = A2.
So, F2 = P2 * A2.
Step 5: Set up the equation using the principles mentioned above.
Since P1 = P2 = P, we can equate the two equations from steps 3 and 4: F1 = F2.
This equation can be written as P1 * A1 = P2 * A2.
Step 6: Solve for the force required to lift the car.
Substituting the values into the equation from step 5:
(P1 * A1) = (P2 * A2)
F1 = F2
P1 * A1 = P2 * A2
P1 = (P2 * A2) / A1
Finally, the force required to lift the car is given by:
F1 = P1 * A1
Substituting the given values:
F1 = [(P2 * A2) / A1] * A1
F1 = P2 * A2
Now, we need to calculate the force exerted on the output piston (F2) using the weight of the car.
Step 7: Find the weight of the car.
The weight of an object is given by the formula: weight = mass * gravity.
Here, the mass of the car is given as 1500 kg, and the acceleration due to gravity is approximately 9.8 m/s^2.
So, weight = 1500 kg * 9.8 m/s^2.
Step 8: Calculate the force exerted on the output piston.
The force exerted on the output piston is equal to the weight of the car.
Therefore, F2 = weight.
Substituting the weight value from step 7, we can find the force exerted on the output piston.
So, the force necessary to lift the 1500-kg car on the output piston with a radius of 0.5 m is equal to the weight of the car, which is calculated in step 7.