Answer:
Explanation:
To calculate the work done by the crane in lifting the wrecking ball, we need to first calculate the gravitational potential energy of the ball before and after the lift, and then find the difference between them.
The gravitational potential energy (U) of an object of mass m at a height h above a reference level is given by the equation:
U = mgh
where g is the acceleration due to gravity, which is approximately 32.2 ft/s^2.
Before the lift, the ball is at ground level, so its height above the reference level is 0 ft. The mass of the ball is 3000 lbs, and we can calculate its initial gravitational potential energy as:
U_initial = (3000 lbs) x (32.2 ft/s^2) x (0 ft) = 0 ft-lbs
After the lift, the ball is 10 ft above the reference level, so its height is h = 10 ft. The length of the cable is also 10 ft, so the ball has been lifted a distance of 10 ft. The mass of the cable can be calculated as:
m_cable = (10 ft) x (7 lb/ft) = 70 lbs
The total mass that the crane has lifted is the sum of the mass of the ball and the mass of the cable:
m_total = 3000 lbs + 70 lbs = 3070 lbs
We can now calculate the final gravitational potential energy of the ball as:
U_final = (3070 lbs) x (32.2 ft/s^2) x (10 ft) = 9,932.4 ft-lbs
The work done by the crane in lifting the ball is the difference between the final and initial gravitational potential energies:
W = U_final - U_initial = 9,932.4 ft-lbs - 0 ft-lbs = 9,932.4 ft-lbs
Therefore, the work done by the crane in lifting the wrecking ball from ground level to 10 ft in the air is 9,932.4 ft-lbs.