Question

You are designing a hydraulic lift for an automobile garage. It will consist of two oil-filled cylindrical pipes of different diameters. A worker pushes down on a piston (with a diameter of 25 cm) at one end, raising the car on a platform at the other end. To handle a full range of jobs, you must be able to lift cars up to 3,471 kg, plus the 451 kg platform on which they are parked. To avoid injury to your workers, the maximum amount of force a worker should need to exert is 106 N. What should be the diameter of the pipe under the platform if the diameter? Use g = 9.8 ™. Express answers in meters (m).

Answer 1

To determine the diameter of the pipe under the platform in the hydraulic lift, we can use Pascal's Law. By equating the pressure exerted by the worker on the piston to the pressure exerted by the weight of the car and platform on the pipe under the platform, we can calculate the diameter. The diameter of the pipe under the platform should be approximately 15.6 cm.

Step-by-step explanation:

To determine the diameter of the pipe under the platform in the hydraulic lift, we need to use the principle of Pascal's Law. According to Pascal's Law, the pressure applied to a fluid confined in a container is transmitted equally in all directions. In this case, we can equate the pressure exerted by the worker on the piston to the pressure exerted by the weight of the car and the platform on the pipe under the platform.

The force exerted by the worker on the piston can be calculated using the formula:
Force = Pressure × Area

Since the diameter of the piston is given as 25 cm, we can calculate its area using the formula:
Area = π × (radius)^2

Now, we can determine the force exerted by the worker on the piston:
Force = Pressure × Area
106 N = Pressure × (π × (12.5 cm)^2)

Next, we can calculate the weight of the car and the platform:
Weight = Mass × Acceleration due to gravity
Weight = (3,471 kg + 451 kg) × 9.8 m/s^2

Finally, we can equate the pressure exerted by the weight of the car and the platform to the pressure exerted by the worker on the piston:
Pressure = Weight / (π × (radius)^2)

By rearranging the equation, we can solve for the diameter of the pipe under the platform:
Diameter = √(Weight / (Pressure × π))

Now, we can substitute the known values and solve for the diameter:

Diameter = √((Weight) / ((106 N / (π × (12.5 cm)^2)) × π))

After performing the calculations, we find that the diameter of the pipe under the platform should be approximately 15.6 cm.

Answer 2

The calculated diameter of the pipe under the platform for the hydraulic lift is approximately 31.2 centimeters (0.312 meters), ensuring efficient lifting of a car and platform with a total weight of 3,922 kg.

Designing the Hydraulic Lift:

Determine the diameter of the pipe under the platform:

1. Calculate the total weight to be lifted:

Weight of car + platform = 3,471 kg + 451 kg = 3,922 kg

2. Convert weight to force:

Force = weight * gravity (g) = 3,922 kg * 9.8 m/s² ≈ 38,423 N

3. Determine the area ratio of the cylinders:

We know the force exerted by the worker (F1) and the maximum force needed to lift the car (F2). We need to find the area ratio (A1/A2) of the pistons such that:

F1/A1 = F2/A2

Substituting the values:

106 N / (π * (25 cm/2)² ) = 38,423 N / A2

4. Solve for the area of the second piston (A2):

Simplifying the equation:

A2 = (π * (25 cm/2)²) * (38,423 N) / 106 N ≈ 844.1 cm²

5. Convert area to diameter of the second piston:

Diameter (d2) = √(4 * A2 / π)

d2 = √(4 * 844.1 cm² / π) ≈ 31.2 cm

Therefore, the diameter of the pipe under the platform should be approximately 31.2 centimeters (0.312 meters).