A piston, A, as shown below, has a diameter of 0.54 cm. A second piston, B has a diameter of 3.8 cm. Determine the force, F, necessary to support the 500.0 N weight in the absence of friction.

To determine the force necessary to support the weight in the absence of friction, we can use Pascal's law, which states that the pressure exerted in a fluid is transmitted uniformly in all directions.

The formula for pressure is:

P = F / A

Where:
P is the pressure exerted,
F is the force applied, and
A is the area over which the force is applied.

We can use this formula to find the force, F, needed to support the weight.

Step 1: Calculate the areas of the pistons.

The area of a circle is given by the formula:

A = π * r^2

Where:
A is the area,
π is a mathematical constant approximately equal to 3.14159, and
r is the radius of the circle.

For piston A:
Radius = diameter / 2 = 0.54 cm / 2 = 0.27 cm.
Area = π * (radius)^2 = 3.14159 * (0.27 cm)^2 = 0.229 cm^2.

For piston B:
Radius = diameter / 2 = 3.8 cm / 2 = 1.9 cm.
Area = π * (radius)^2 = 3.14159 * (1.9 cm)^2 = 11.305 cm^2.

Step 2: Calculate the pressure exerted on each piston.

Since the pressure is transmitted uniformly, the pressure exerted on piston A is the same as the pressure exerted on the weight, which is the same as the pressure exerted on piston B.

Let's say the pressure exerted is P.

Pressure on piston A = P = F / A_A = 500 N / 0.229 cm^2
Pressure on piston B = P = F / A_B = 500 N / 11.305 cm^2

Step 3: Equate the pressures on the pistons.

Since the pressure exerted is the same on both pistons:

F / A_A = F / A_B

Step 4: Solve for the force, F.

Cross multiply:

F * A_B = F * A_A

Divide by F:

A_B = A_A

Substitute the values of A_A and A_B:

11.305 cm^2 = 0.229 cm^2

Step 5: Solve for F.

F = 500 N * 11.305 cm^2 / 0.229 cm^2

F = 24544 N

Therefore, the force necessary to support the 500.0 N weight in the absence of friction is approximately 24544 N.

To determine the force necessary to support the weight in the absence of friction, we can use the principle of Pascal's law, which states that the pressure applied to a fluid in a confined space is transmitted equally in all directions.

First, we need to calculate the areas of pistons A and B.

The formula for the area of a circle is:

Area = π * (radius)^2

Given that the diameter of piston A is 0.54 cm, the radius is half of the diameter:

Radius of piston A = 0.54 cm / 2 = 0.27 cm

Area of piston A = π * (0.27 cm)^2

Similarly, for piston B, the diameter is 3.8 cm:

Radius of piston B = 3.8 cm / 2 = 1.9 cm

Area of piston B = π * (1.9 cm)^2

Next, we can calculate the pressure on each piston.

The formula for pressure is:

Pressure = Force / Area

Since the pressure is transmitted equally in all directions, the pressure on piston A must be the same as the pressure on piston B.

Given that the weight is 500.0 N and there is no friction, the force required to support the weight is equal to the weight itself.

So, for piston A, the pressure can be calculated:

Pressure on piston A = Force on piston A / Area of piston A

Since the pressure is the same on both pistons, we can set up an equation:

Pressure on piston A = Pressure on piston B

Force on piston A / Area of piston A = Force on piston B / Area of piston B

Substituting the values:

500.0 N / Area of piston A = Force on piston B / Area of piston B

We can rearrange the equation to solve for the force on piston B:

Force on piston B = (500.0 N / Area of piston A) * Area of piston B

Substituting the values for the areas:

Force on piston B = (500.0 N / (π * (0.27 cm)^2)) * (π * (1.9 cm)^2)

Simplifying the equation:

Force on piston B ≈ 2,453.7 N

Therefore, the force necessary to support the 500.0 N weight in the absence of friction is approximately 2,453.7 Newtons.

pressure is the same.

Fa/areaA=Fb/AreaB

area is proportional to diamter^2