Conveyor Drive Mechanical Work

In automated logistics facilities, motorized conveyor systems pull cargo carts along linear tracks. Mechanical engineers calculate the mechanical work done by tension cables to size drive motors and optimize electrical energy consumption.

A motor-driven cable pulls a cargo pallet along a straight horizontal rail with displacement vector $\vec{d} = 12\hat{i}\text{ meters}$. Because of the overhead placement of the drive pulley, the cable pulls on the pallet with a constant force vector $\vec{F} = 350\hat{i} + 120\hat{j}\text{ Newtons}$.

1. Compute the mechanical work done by the cable on the pallet using the dot product.
2. Find the angle $\theta$ between the force vector and the direction of motion.
3. Determine what portion of the total force magnitude contributes directly to moving the pallet forward.