piston 1 has a diameter of 0.25 in; pistol 2 has a diameter of 1.5 in. In the absence of friction, determine the force necessary to support the 500 lb weight.
I will have to guess what your problem is because if you tried to copy and paste a picture it did not work.
If we are talking a hydraulic problem with force on the small piston lifting a weight on the big one then
F =(little area/big area) weight
F = (.25^2/1.5^2) 500
F = [ 1/(4*2.25) ] 500
F = 55.56 lb
To determine the force necessary to support the 500 lb weight, we need to consider the principles of fluid mechanics and Pascal's Law.
Pascal's Law states that when there is an enclosed fluid in a container, any change in pressure applied to the fluid is transmitted undiminished to all portions of the fluid and to the walls of the container.
In this case, we can consider the fluid as the piston. The force exerted by the fluid on the larger piston (piston 2) will be equal to the force exerted by the smaller piston (piston 1) which is supporting the weight.
First, we need to calculate the areas of the pistons. The area of a circle is calculated using the formula:
Area = π * (diameter/2)^2
Let's calculate the areas of the pistons:
Area1 = π * (0.25/2)^2 = 0.049 sq. in.
Area2 = π * (1.5/2)^2 = 1.767 sq. in.
Now, we can use Pascal's Law to find the force necessary to support the weight.
Pressure1 = Pressure2
The pressure exerted by a fluid is calculated using the formula:
Pressure = Force/Area
Since the pressure is the same for both pistons:
Force1/Area1 = Force2/Area2
Rearranging the equation to solve for Force2:
Force2 = (Force1/Area1) * Area2
Given that Force1 is 500 lb, let's calculate Force2:
Force2 = (500 lb / 0.049 sq. in.) * 1.767 sq. in.
Calculating this expression, we find that the force necessary to support the 500 lb weight is approximately 9,082 lb.
Therefore, the force required to support the 500 lb weight is approximately 9,082 lb in the absence of friction.
To determine the force necessary to support the 500 lb weight, we can use Pascal's law, which states that pressure applied to an enclosed fluid is transmitted equally in all directions.
Step 1: Calculate the pressure at the smaller piston (piston 1).
- Given: Piston 1 diameter (d₁) = 0.25 in
- Calculate the radius of piston 1 (r₁) = d₁/2 = 0.25/2 = 0.125 in
- Convert the radius to inches (r₁_inches) by multiplying with π = 0.125 * π = 0.3927 in
- Calculate the area of piston 1 (A₁) = π * (r₁_inches)^2 = π * (0.3927)^2 = 0.4843 in^2
- Calculate the pressure at piston 1 (P₁) = Force (F₁) / Area (A₁)
Step 2: Calculate the pressure at the larger piston (piston 2).
- Given: Piston 2 diameter (d₂) = 1.5 in
- Calculate the radius of piston 2 (r₂) = d₂/2 = 1.5/2 = 0.75 in
- Convert the radius to inches (r₂_inches) by multiplying with π = 0.75 * π = 2.3562 in
- Calculate the area of piston 2 (A₂) = π * (r₂_inches)^2 = π * (2.3562)^2 = 17.671 in^2
Step 3: Calculate the force required to support the weight (F₂).
- Given: Weight (W) = 500 lb
- Since pressure is transmitted equally, the pressure at piston 2 (P₂) = P₁ (pressure at piston 1)
- P₂ = P₁ = F₂ / A₂ (from Pascal's law)
- Rearrange the equation to solve for F₂: F₂ = P₂ * A₂
Let's perform the calculations now:
Step 1:
- r₁_inches = 0.125 in
- A₁ = 0.4843 in^2
- P₁ = F₁ / A₁
Step 2:
- r₂_inches = 0.75 in
- A₂ = 17.671 in^2
Step 3:
- W = 500 lb
- P₂ = P₁
- F₂ = P₂ * A₂