Diameter of Football Players' Piston: Calculating

[pshla]

Homework Statement

A 55 kg cheerleader uses an oil-filled hydraulic lift to hold four 110 kg football players at a height of 1.0m. If her piston is 16 cm in diameter, what is the diameter of the football players' piston?

Homework Equations

F1/A1=F2/A2
A=pi*r^2
F=mg

The Attempt at a Solution

(55)(9.8)/pi(0.08)^2=(440)(9.8)/pi(r^2)

After I did this, I realized that I need to work the height into the equation, but I'm not sure where it would fit in.

 

[mgb_phys]

The simplest way is to think of the volume displaced - since the hydraulic fluid is in incompressible that must be the same for the cheerleader and the footballers.
In addition the force exerted by the cheerleader and footballers is known.

area_cheerleader * height_cheer = area_footballers * height_footballers
55g / area_chearleader = 110g/area_footballer

The unknowns are area_footballer and height_cheerleader.

 

[m2003]

To calculate the diameter of the football players' piston, we can use the equation F1/A1=F2/A2, where F1 and A1 are the force and cross-sectional area of the cheerleader's piston, and F2 and A2 are the force and cross-sectional area of the football players' piston. We can also use the equation F=mg to represent the force of gravity acting on the football players and the cheerleader.

First, we need to calculate the force of gravity acting on the football players. Since they have a combined mass of 440 kg, the force of gravity can be calculated as F=mg=(440)(9.8)=4312 N.

Next, we can calculate the cross-sectional area of the cheerleader's piston, using the equation A=pi*r^2. Since her piston has a diameter of 16 cm, the radius would be 8 cm or 0.08 m. Thus, the cross-sectional area of her piston is A=pi*(0.08)^2=0.0201 m^2.

Now, we can plug in these values into the equation F1/A1=F2/A2 and solve for A2, which represents the cross-sectional area of the football players' piston. This gives us A2=(F1/A1)*A2=(4312 N/0.0201 m^2)=214678.6 m^2.

Finally, we can use the equation A=pi*r^2 to solve for the radius of the football players' piston. Plugging in the calculated value for A2, we get 214678.6 m^2=pi*r^2. Solving for r, we get r=260.4 m. Therefore, the diameter of the football players' piston would be approximately 520.8 m.

It's important to note that this is a very large diameter and may not be feasible for a football player's piston. It's possible that the piston would have a smaller diameter, but a larger force acting on it to lift the football players. Additionally, the height of 1.0m may also play a role in determining the appropriate diameter for the football players' piston. Further research and calculations would be needed to determine a more accurate and feasible diameter for their piston.