The crane shown in the figure is lifting a 363-kg crate upward with an acceleration of 3.03 m/s². The cable from the crate passes over a solid cylindrical pulley at the top of the boom. The pulley has a mass of 155 kg. The cable is then wound onto a hollow cylindrical drum that is mounted on the deck of the crane. The mass of the drum is 187 kg, and its radius (the same as that of the pulley) is 0.693 m. The engine applies a counterclockwise torque to the drum in order to wind up the cable.

What is the magnitude of this torque? Ignore the mass of the cable.

First, let's calculate the moment of inertia of the drum. Since the drum is hollow, its moment of inertia can be expressed as the difference between the moment of inertia of the outer cylinder and the moment of inertia of the inner cylinder.

The moment of inertia of a solid cylinder is given by:

[tex]I_{solid}[/tex] = (1/2) * mass * [tex]\rm radius^2[/tex]

The moment of inertia of the hollow cylinder (the drum) is:

[tex]I_{drum} = I_{outer} - I_{inner}[/tex]

The moment of inertia of the pulley is:

[tex]I_{pulley} = (1/2) * mass_{pulley} * radius_{pulley^2}[/tex]

Now, we can calculate the moment of inertia of the drum:

[tex]I_{drum} = (1/2) * mass_{drum} * radius^2 - I_{pulley}[/tex]

Next, we calculate the torque:

Torque = [tex]I_{drum}[/tex] * Angular acceleration

Substituting the given values:

[tex]\rm Torque = (1/2) * 187 kg * (0.693 m)^2 - (1/2) * 155 kg * (0.693 m)^2 * 3.03 m/s^2[/tex]

Calculating this expression gives a magnitude of approximately 2587.61 Nm.

Therefore, the magnitude of the torque applied by the engine to wind up the cable is 2587.61 Nm.

Know more about torque:

https://brainly.com/question/28220969

#SPJ12