A hydraulic press has an input cylinder 1 m in diameter and an output cylinder 2 meters in diameter. Assuming 100% efficiency, find the force exerted by the output piston when a force of 10 Newtons is applied to the input piston.
pounds
If the input piston is moved through 5 meters, how far is the output piston moved?
meters
To find the force exerted by the output piston, we can use the formula for hydraulic press:
(Area of input piston) / (Area of output piston) = (Force exerted by input piston) / (Force exerted by output piston)
Given that the diameter of the input piston is 1 meter, the radius is 0.5 meters. Therefore, the area of the input piston is:
Area of input piston = π × (0.5)^2 = π × 0.25 ≈ 0.7854 square meters
Given that the diameter of the output piston is 2 meters, the radius is 1 meter. Therefore, the area of the output piston is:
Area of output piston = π × (1)^2 = π × 1 ≈ 3.1416 square meters
Now, if a force of 10 Newtons is applied to the input piston, we can rearrange the formula for hydraulic press to find the force exerted by the output piston:
Force exerted by output piston = (Force exerted by input piston) × (Area of output piston) / (Area of input piston)
Force exerted by output piston = 10 × 3.1416 / 0.7854
Force exerted by output piston ≈ 40 Newtons
Therefore, the force exerted by the output piston is approximately 40 Newtons.
To find how far the output piston is moved, we can use the concept of equal pressures in a hydraulic system. According to Pascal's law, the pressure exerted at any point in a confined fluid is transmitted equally in all directions. In the case of a hydraulic press, the pressure at the input piston is the same as the pressure at the output piston.
Pressure = Force / Area
Therefore, the pressure at the input piston is:
Pressure = 10 / 0.7854 ≈ 12.732 N/m^2
Since the pressure at the output piston is the same, we can find the distance moved by the output piston using the formula:
Distance moved by output piston = (Force exerted by output piston) / (Pressure at output piston)
Distance moved by output piston = 40 / 12.732 ≈ 3.1416 meters
Therefore, when the input piston is moved through 5 meters, the output piston is moved approximately 3.1416 meters.
To find the force exerted by the output piston, we can use Pascal's law, which states that the pressure applied to a fluid in a closed system is transmitted equally in all directions. In our case, the pressure on the input piston (Given as 10 Newtons) will be transmitted to the output piston.
First, we need to find the area of both pistons. The area of a circle is given by the formula A = πr^2, where r is the radius of the circle.
For the input piston:
Radius = Diameter/2 = 1 m/2 = 0.5 m
Area (A1) = π(0.5^2) = 0.7854 m^2
For the output piston:
Radius = Diameter/2 = 2 m/2 = 1 m
Area (A2) = π(1^2) = 3.1416 m^2
Since pressure (P) is force (F) divided by area (A), and the pressure on both pistons must be the same due to Pascal's law, we can set up the following equation:
P1 = P2
F1/A1 = F2/A2
Plugging in the given force on the input piston (F1 = 10 Newtons) and the areas we calculated, we can solve for the force on the output piston (F2):
10/0.7854 = F2/3.1416
F2 = (10/0.7854) * 3.1416
F2 ≈ 39.9 Newtons
Therefore, the force exerted by the output piston is approximately 39.9 Newtons.
Now, let's move on to the second part of the question.
To find how far the output piston is moved when the input piston is moved through 5 meters, we can use the principle of equalizing the volumes of the fluid in the hydraulic press.
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius of the cylinder and h is the height (distance traveled).
For the input piston:
Radius = Diameter/2 = 1 m/2 = 0.5 m
Distance (h1) = 5 m
Volume (V1) = π(0.5^2)(5) ≈ 3.926 m^3
For the output piston:
Radius = Diameter/2 = 2 m/2 = 1 m
Volume (V2) = π(1^2)(x) = π(x), where x is the unknown distance the output piston is moved.
Since the volume of the fluid is the same in a closed system, we can set up the following equation:
V1 = V2
3.926 = π(x)
Solving for x:
x = 3.926/π
x ≈ 1.25 meters
Therefore, the output piston is moved approximately 1.25 meters when the input piston is moved through 5 meters.
F2/F1 = S2/S1 =πR^2/πr^2= (R/r)^2
S1=V1/h1
S2=V2/h2
F2/F1 =h1/h2