A hydraulic press has an input cylinder 4.0cm in diameter and an output cylinder 16cm in diameter. (a) Assuming 100 percent efficiency, find the force exerted by the output piston when a force of 170N is applied to the input piston.

Pressure is the same, and the area of the output is proportional to the ratio of diamenters squared, so

output force=(16/4)^2 *170N
= 16*170N

To solve this problem, we will use Pascal's law, which states that pressure is transmitted equally in a closed fluid system. We can use this law to determine the relationship between the forces exerted on the input and output pistons.

Step 1: Calculate the ratio between the areas of the input and output pistons:
The area of a circle can be calculated using the formula: A = π r^2.
Given that the diameter of the input piston is 4.0 cm, the radius (r) can be calculated by dividing the diameter by 2: r = d/2 = 4.0 cm / 2 = 2.0 cm = 0.02 m.
So, the area of the input piston (A_input) is: A_input = π (0.02 m)^2.

Similarly, the diameter of the output piston is 16 cm, so the radius (r) of the output piston is: r = d/2 = 16 cm / 2 = 8 cm = 0.08 m.
Therefore, the area of the output piston (A_output) is: A_output = π (0.08 m)^2.

Step 2: Calculate the force exerted by the output piston:
According to Pascal's law, the pressure is equal in both pistons. So, we can equate the pressure on the input piston to the pressure on the output piston:
P_input = P_output,
where P_input = F_input / A_input and P_output = F_output / A_output.

Given that 100% efficiency is assumed, the force exerted by the output piston is the same as the applied force on the input piston.
So, F_input = F_output = 170 N.

Using the relationship between pressure and force, we can rewrite the equation as:
F_input / A_input = F_output / A_output.

Substituting the known values, we have:
170 N / A_input = 170 N / A_output.

Step 3: Calculate the force exerted by the output piston:
Now, we substitute the values of A_input and A_output into the equation to calculate the force exerted by the output piston.

A_input = π (0.02 m)^2 = 0.00126 m^2,
A_output = π (0.08 m)^2 = 0.02011 m^2.

Substituting these values into the equation, we have:
170 N / 0.00126 m^2 = F_output / 0.02011 m^2.

To find F_output, we rearrange the equation and solve for F_output:
F_output = (170 N / 0.00126 m^2) * 0.02011 m^2.

Using a calculator, we can solve this equation to find the force exerted by the output piston.

By following these steps, we can determine the force exerted by the output piston using the given values and utilizing Pascal's law.