The mechanism shown in the figure (Figure 1) is used to raise a crate of supplie

Question

The mechanism shown in the figure (Figure 1) is used to raise a crate of supplies from a ship's hold. The crate has total mass 51 kg . A rope is wrapped around a wooden cylinder that turns on a metal axle. The cylinder has radius 0.20 m and a moment of inertia I = 3.0 kgm2 about the axle. The crate is suspended from the free end of the rope. One end of the axle pivots on frictionless bearings; a crank handle is attached to the other end. When the crank is turned, the end of the handle rotates about the axle in a vertical circle of radius 0.12 m , the cylinder turns, and the crate is raised.

What magnitude of the force F  applied tangentially to the rotating crank is required to raise the crate with an acceleration of 1.40 m/s2 ? (You can ignore the mass of the rope as well as the moments of inertia of the axle and the crank.)

Express your answer using two significant figures.

Explanation / Answer

Force required to accelerate the crate, Fc = weight (mg) + accel. force (ma)
Fc = 51kg x (9.81 + 1.40)m/s = 571.71N
Torque at cylinder due to crate, Tc = Fc x r = 571.71N x 0.20m = 114.342 Nm

Torque required to accel. cylinder, T' = I. .. (MoI x ang.accel. .. equivalent to F = ma)
T' = 3.0kg.m² x (1.40 / 0.20) = 21 Nm .. (v = rdv/dt = r.d/dt a = r. = a/r)

Total torque required = Tc + T' = (114.342 + 21) = 135.342 Nm

Torque at handle, Fh x 0.12m = 135.342 Nm

Fh = 135.342 / 0.12 .. ..

Fh = 1127.85 N

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