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 56 kg. A rope is wrapped around a wooden cylinder that turns on a metal axle. The cylinder has radius 0.28 m and a moment of inertia I = 2.6 kg middot m^2 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 vector F applied tangentially to the rotating crank is required to raise the crate with an acceleration of 1.40 million/seconds^2 ? (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 = (mg) + (ma)
Fc = 56kg x (9.80 + 1.40)m/s = 627.2 N

Torque at cylinder due to crate, Tc = Fc x r = 627.2N x 0.28m = 175.616 Nm

Torque required to accel. cylinder, T' = I... (MoI x ang.accel. .. equivalent to F = ma)
T' = 2.60kg.m² x (1.40 / 0.28) = 13Nm

Total torque required = Tc + T' = (175.616 +13) = 188.616 Nm

Torque at handle, Fh x 0.12m = 188.616 Nm

Fh = 188.616 / 0.12 = 1571.8 N

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