The crane shown in the drawing is lifting a 170-kg crate upward with an acceleration of 1.1 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 130 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 150 kg, and its radius is 0.76 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.
So far I have T=Ialpha and I am stuck on how to approach this problem.
To solve this problem, you can use the equation τ = Iα, where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
1. Find the moment of inertia of the drum:
- The moment of inertia of a hollow cylindrical drum is given by I = (1/2)mr^2, where m is the mass and r is the radius.
- Substituting the given values, the moment of inertia of the drum is I_drum = (1/2)(150 kg)(0.76 m)^2.
2. Find the moment of inertia of the pulley:
- The moment of inertia of a solid cylindrical pulley is given by I = (1/2)mr^2.
- Substituting the given values, the moment of inertia of the pulley is I_pulley = (1/2)(130 kg)(0.76 m)^2.
3. Find the total moment of inertia:
- Since the pulley and the drum are rotating together, you need to add their individual moments of inertia to get the total moment of inertia.
- The total moment of inertia is I_total = I_drum + I_pulley.
4. Find the angular acceleration:
- The angular acceleration (α) is given as 1.1 m/s^2.
5. Calculate the torque:
- Use the equation τ = Iα and substitute the values of I_total and α.
- The torque exerted by the engine is τ = I_total * α.
By following these steps, you should be able to find the magnitude of the torque exerted by the engine to wind up the cable.