A 57.0 cheerleader uses an oil-filled hydraulic lift to hold four 110 football players at a height of 1.40 . If her piston is 19.0 in diameter, what is the diameter of the football players' piston? The density of the oil in the hydraulic system is 900 kg/m^3.
Force1*area1=force2*area2
Cherleader *19^2=4*110*d^2
Not having any units given, I can't go farther. Solve for d
To find the diameter of the football players' piston, we can use the principle of Pascal's law, which states that pressure applied to an enclosed fluid is transmitted uniformly in all directions.
First, let's find the force exerted by the cheerleader on the hydraulic lift. The force is given by the product of mass and gravitational acceleration (F = m * g). The mass of the cheerleader is not given, so we cannot calculate the exact force. However, it is not necessary to know the actual force in this problem.
Next, we can use Pascal's law to equate the pressure in both pistons of the hydraulic system. The formula is P1/A1 = P2/A2, where P1 and P2 are the pressures in the cheerleader's piston and football players' piston, respectively, and A1 and A2 are the areas of the respective pistons.
Let's denote the diameter of the football players' piston as D2. The area of a piston is calculated using the formula A = π * (D/2)^2, where D is the diameter. Substituting in this formula and rearranging the equation, we get:
P1 / (π * (D1/2)^2) = P2 / (π * (D2/2)^2)
Since we know the diameter of the cheerleader's piston (D1 = 19.0 in), we can plug in the values and solve for D2.
Let's convert the diameter of the cheerleader's piston to meters since the density of the oil is given in kg/m^3. 1 inch is approximately equal to 0.0254 meters.
D1 = 19.0 in * 0.0254 m/in = 0.4826 m
Now, let's plug the values into the equation:
P1 / (π * (0.4826/2)^2) = P2 / (π * (D2/2)^2)
Since the pressure exerted by the hydraulic lift depends on the weight of the players and the area of the cheerleader's piston, we can consider it as a constant, denoted as C.
C = P1 / (π * (0.4826/2)^2)
With this simplification, the equation becomes:
C = P2 / (π * (D2/2)^2)
We can rearrange for D2:
D2/2 = sqrt(P2 / (C * π))
D2 = 2 * sqrt(P2 / (C * π))
Now, let's calculate D2. We will need the pressure P2, which is related to the weight of the players and the area of their piston. The weight is given by the product of mass and gravitational acceleration (W = m * g). The mass of each player is not given, so we cannot calculate the exact weight. However, it is not necessary to know the actual weight in this problem.
Finally, let's substitute all the known values and solve for D2:
D2 = 2 * sqrt(P2 / (C * π))
Note: Please ensure that all the units are consistent throughout the calculations.