A 55.0kg cheerleader uses an oil-filled hydraulic lift to hold four 130kg football players at a height of 1.20m. If her piston is 19.0cm in diameter, what is the diameter of the football players' piston?

To find the diameter of the football players' piston, we can use the principle of Pascal's law, which states that pressure applied to a fluid in a closed system is transmitted equally in all directions.

Let's start by determining the pressure exerted on the cheerleader's piston. We know that the weight of the cheerleader is equal to the force exerted by the lift on her. The force can be calculated using the formula:

Force = mass * acceleration due to gravity

Force = 55.0 kg * 9.8 m/s^2 = 539 N

Since both the cheerleader and the football players are at the same height, the pressure exerted by the cheerleader's piston is equal to the pressure exerted by the football players' piston. The pressure can be calculated using the formula:

Pressure = Force / Area

The area of the cheerleader's piston can be calculated using the formula for the area of a circle:

Area = π * radius^2

Since we are given the diameter of the cheerleader's piston, we need to divide it by 2 to get the radius.

Radius = 19.0 cm / 2 = 9.5 cm = 0.095 m

Area = π * (0.095 m)^2 = 0.02824 m^2

Now, we can calculate the pressure exerted by the cheerleader's piston:

Pressure = 539 N / 0.02824 m^2 = 19073.66 Pa

Since the pressure is the same for both pistons, we can use the following formula to find the diameter of the football players' piston:

Diameter = 2 * square root of (Pressure / (π * pressure on football players))

The pressure on the football players can be calculated by multiplying the weight of all four players by the acceleration due to gravity:

Pressure on football players = (mass of football players * acceleration due to gravity) * 4

Mass of football players = 130 kg * 4 = 520 kg

Pressure on football players = (520 kg * 9.8 m/s^2) * 4 = 20272 Pa

Substituting the values into the formula, we get:

Diameter = 2 * square root of (19073.66 Pa / (π * 20272 Pa))

Diameter = 2 * square root of (0.2967)

Diameter ≈ 2 * 0.5443

Diameter ≈ 1.0886 m

Therefore, the diameter of the football players' piston is approximately 1.0886 meters.