Piston Circle 1 in the figure below has a diameter of 0.29 in.; Piston Circle 2 has a diameter of 1.7 in. In the absence of friction, determine the force vector F necessary to support the 500-lb weight.
To determine the force vector F necessary to support the 500-lb weight, we need to consider the balance of forces acting on the system.
First, let's calculate the areas of the two circles:
Area of Piston Circle 1:
A1 = π * (diameter of Circle 1/2)^2
= π * (0.29 in/2)^2
= π * (0.145 in)^2
≈ 0.066 in^2
Area of Piston Circle 2:
A2 = π * (diameter of Circle 2/2)^2
= π * (1.7 in/2)^2
= π * (0.85 in)^2
≈ 2.268 in^2
Next, let's calculate the pressure exerted on the fluid in each circle:
Pressure on Piston Circle 1:
P1 = force on Piston Circle 1 / A1
Pressure on Piston Circle 2:
P2 = force on Piston Circle 2 / A2
Since the fluid pressure is the same throughout the system in the absence of friction, we can set P1 equal to P2:
P1 = P2
Dividing both sides by π:
(force on Piston Circle 1 / A1) = (force on Piston Circle 2 / A2)
Rearranging the equation to solve for the force on Piston Circle 1:
force on Piston Circle 1 = (P1 * A1) / A2 * force on Piston Circle 2
Given that the 500-lb weight exerts a downward force on Piston Circle 2, we convert it to force in pounds per square inch (psi):
Force on Piston Circle 2 = 500 lb
Converting this force to psi (assuming the weight is evenly distributed over the Piston Circle 2 area):
Force on Piston Circle 2 = 500 lb / A2
Finally, substituting the values into the equation:
force on Piston Circle 1 = (P1 * A1) / A2 * (500 lb / A2)
Now, you can evaluate the above expression substituting the values of P1, A1, and A2 to calculate the force vector F necessary to support the 500-lb weight.