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Answer :
The magnitude of the torque applied by the engine to wind up the cable is 2587.61 Nm.
To calculate the magnitude of the torque applied by the engine to wind up the cable, we need to consider the rotational dynamics of the system.
The torque can be calculated using the formula:
Torque = Moment of inertia * Angular acceleration
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.
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