A loaded ore car has a mass of 950 kg and rolls on rails with negligible frictio

Question

A loaded ore car has a mass of 950 kg and rolls on rails with negligible friction. It starts from rest and is pulled up a mine shaft by a cable connected to a winch. The shaft is inclined at 33.5° above the horizontal. The car accelerates uniformly to a speed of 2.25 m/s in 13.0 s and then continues at constant speed.

(a) What power must the winch motor provide when the car is moving at constant speed?
kW

(b) What maximum power must the motor provide?
kW

(c) What total energy transfers out of the motor by work by the time the car moves off the end of the track, which is of length 1,250 m?
J

Explanation / Answer

force required to move the car up the rails F = mg * sin(33.5)

force required to move the car up the rails F = 950 * 9.8* sin(33.5)

F = 5138.53 Newtons

once it reaches constant speed,

in one second, it moves 2.25 meters

so,

work = 5138.53 * 2.25

work = 11561.6925 Joules

since this is one second, that is a rate of 11561.6925 watts (a)

While accelerating, it must provide additional work.

average acceleration = 2.25/13

average acceleration = 0.173 m/s²

distance = 2.25 m/s x (1/2) x 13s

distance = 14.625 m

F = ma

F = 950 * 0.173

F = 164.35 Newtons

The power to do this is 164.35 * 14.625 /13

The power to do this is = 184.893 watts

total power = 11561.6925 + 184.893

total power = 11746.5855 watts (b)

1250m x 5138.53 N = 6423162.5 J

total energy transfers out of the motor by work by the time the car moves off the end of the track, which is of length 1,250 m = 6423162.5 J (c)

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