Need Help With Physics Problem Involving Work and Power?

Question

You are asked to design a construction crane that is capable of lifting a mass of 5,000 kg (5 metric tons) from the ground to the top of a 65 m high building in 30 seconds. What power should you specify for the motor that lifts the load, in horsepower? Note 1 hp = 746 Watts = 746 Joules/second.

Average power is defined to be the work done by a force divided by the time it took to do that work,

P_avg = W/Delta(t)

Instantaneous power is the rate at which you are doing work

P = dW/dt = F * dx /dt = F * v,

where dx, F and v are the infinitesimal displacement vector, the force vector, and the velocity vector, dt is the infinitesimal elapsed time, and the "." above is the dot (or scalar) product of the vectors.

Answer 1

E/t = P the power required; E = mgh; where m = 5000 kg, g = 9.81 m/sec^2, and h = 65 m.  t = 30 sec.  You can do the math.

The physics is this, E is just the work against gravity to lift the mass m and height h.  That work has to be done within t seconds.  Instantaneous power is not germane here.  I'm not clear if such a thing has any meaning; after all, power is energy working over some interval of time.  If that interval is very very tiny, what has that energy over time accomplished?

Answer 2

Power = P =  Work/Time

P = mgh/t

where

m = mass to be lifted =5000 kg (given)

g = acceleration due to gravity = 9.8 m/sec (constant)

h = lifting height = 65 m

t = 30 sec.

Substituting values,

Power = (5000)(9.8)(65)/30

Power = 106,166.7 joules/sec. = 106,166.7 watts

Power = 106,166.7/746

Power = 142.3 hp.

Assuming a crane efficiency of, say, 75%, then the motor must be

142.3/0.75 = 189.75 hp.

NOTE: I would recommend a 200-hp motor for this operation.