A simple model of a crane involves a motor driven drum that winds or unwinds a s

Question

A simple model of a crane involves a motor driven drum that winds or unwinds a steel cable onto the drum. Refer to picture. The cable passes through a pair of guides and over a pulley at the end of the crane arm. Ignore the mass of the cable in the calculations. The drum has a mass of 650 kg and a 1.4 m radius. The pulley has a radius of 0.85 m and a mass of 295 kg. The load has a mass of 1800 kg.

a) If the load is to be accelerated from rest to a constant lifting speed of 30 cm/s over a distance of 0.10 m, what minimum torque must the motor provide?
b) What minimum amount of energy is required by the motor to lift the load from rest to a final height of 10 m where it ends at rest?
c)What are the tensions in the section of cable from the drum to the pulley, and from the pulley to the load as the load is accelerated? What are the tensions when the load is lifted at constant speed?

Explanation / Answer

a). v^2 -u^2 =2a*s => 0.3^2 -0 = 2a*0.1 => a = 0.45 m/s2 angular acceleration of drum = a/r = 0.45/1.4 = 0.32 rad/s torque = I* angular acceleration = (650*1.4^2/2) *0.32 = 203.84 N-m b). energy = change in Gravitational P.E. = 1800*9.8*10 = 176400 J c). angular acceleration of pulley = 0.45/0.85 = 0.529 rad/s let tension in the cable from load to pulley be T1 and that from pulley to drum be T2. for load; T1 -1800 g = 1800*a => T1 = 18450 N for drum; T2 *1.4 = (650*1.4^2/2)*0.32 => T2 = 145.6 N when the load is moving with constant speed, a =0 so, T1 = 1800*9.8 = 17640 N and T2 = T1 when angular acceleration is 0, so T2 = 145.6 N

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