a hydraulic lift is designed for a gain of 100, so that a 10 N force applied at theinput piston will produce a force of 1,000 N at the output piston. If the radius of the input piston is 2 cm, the radius of the output is
a) 200 cm,
b) .02 cm
c) 400
d) 20
e) .05
area out / area in = 100 = pi Rout^2/ pi Rin^2 = R^2/(2^2)
R^2 = 400
R = 20
so ten times the radius for 100 times the area of course :)
We can use the formula for hydraulic lift gain to find the radius of the output piston.
The gain of a hydraulic lift is given by the formula:
Gain = (output force) / (input force)
In this case, the gain is 100, the input force is 10 N, and the output force is 1,000 N. So we have:
100 = 1,000 N / 10 N
Simplifying the equation, we find:
100 = 100
Now, let's consider the hydraulic lift gain formula in terms of the radii of the pistons:
Gain = (output piston radius)^2 / (input piston radius)^2
We can rearrange the formula to solve for the output piston radius:
(output piston radius)^2 = Gain * (input piston radius)^2
Substituting the known values, we have:
(output piston radius)^2 = 100 * (0.02 m)^2
Now, let's calculate the output piston radius:
(output piston radius)^2 = 100 * (0.02^2)
(output piston radius)^2 = 100 * 0.0004
(output piston radius)^2 = 0.04
Taking the square root of both sides, we find:
output piston radius = √0.04
Now, let's calculate this:
output piston radius = 0.2
Therefore, the radius of the output piston is 0.2 m or 20 cm.
Answer: d) 20
To determine the radius of the output piston, we can use the formula for hydraulic gain:
Hydraulic Gain = (Area of output piston) / (Area of input piston)
The force applied at the input piston, F1 = 10 N, and the force produced at the output piston, F2 = 1000 N.
Given that the radius of the input piston is 2 cm, we can calculate its area, A1, using the formula:
A1 = π * (radius of input piston)^2
Substituting the values, we get:
A1 = π * (2 cm)^2 = 4π cm^2
Similarly, we can calculate the area of the output piston, A2, using the formula:
A2 = π * (radius of output piston)^2
Now, let's rearrange the hydraulic gain formula to solve for the radius of the output piston:
Hydraulic Gain = (Area of output piston) / (Area of input piston)
100 = A2 / A1
Substituting the values, we get:
100 = (π * (radius of output piston)^2) / (4π cm^2)
Now, we can cancel out the π terms from both sides of the equation:
100 = (radius of output piston)^2 / 4 cm^2
Multiplying both sides by 4, we get:
400 = (radius of output piston)^2
Taking the square root of both sides, we find:
radius of output piston = √400 cm = 20 cm
Therefore, the radius of the output piston is 20 cm. Thus, the answer is option d) 20.