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 45 kg. A rope is wrapped around a wooden cylinder that turns on a metal axle. The cylinder has radius 0.34 m and a moment of inertia I = 2.9 kgmiddotm^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 F ^rightarrow applied tangentially to the rotating crank is required to raise the crate with an acceleration of 1.40 m/s^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

The handle radius is rather small .. smaller than the winding cylinder !!


Force required to accelerate the crate, Fc = weight (mg) + accel. force (ma)
Fc = 45kg x (9.80 + 1.40)m/s = 504N
Torque at cylinder due to crate, Tc = Fc x r = 504N x 0.34m = 171.36 Nm

Torque required to accel. cylinder, T' = I. .. (MoI x ang.accel. .. equivalent to F = ma)
T' = 2.9kg.m² x (1.40 / 0.34) = 11.94 Nm .. (v = rdv/dt = r.d/dt a = r. = a/r)

Total torque required = Tc + T' = (171.36 + 11.94) = 183.3 Nm

Torque at handle, Fh x 0.12m = 183.3 Nm .. .. Fh = 183.3 / 0.12 .. .. Fh = 1527.5 N

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