The Overloaded Rope
A team is using a rope and pulley system to lift a 200 kg crate from the ground to a platform 5 m high. The rope has a breaking strength of 2500 N. The pulley system has a mechanical advantage of 2 (i.e., the effort force required is half the load). However, due to friction, the efficiency is 80%. Acceleration due to gravity is 9.8 m/s². The team can exert a maximum effort of 1200 N. The crate must be lifted at a constant speed.
1. Calculate the actual effort force required to lift the crate at constant speed, and determine if the rope will break.
Read the worked answer
Load force = mass * g = 200 * 9.8 = 1960 N. Ideal mechanical advantage = 2, so ideal effort = 1960 / 2 = 980 N. With efficiency 80%, actual effort = 980 / 0.8 = 1225 N. The rope breaking strength is 2500 N. The tension in the rope is equal to the effort force (1225 N) in this simple pulley system? Actually, in a pulley system with mechanical advantage 2, the tension in the rope is equal to the effort force. So tension = 1225 N, which is less than 2500 N, so the rope will not break. However, the effort required (1225 N) exceeds the team's maximum effort (1200 N), so they cannot lift it.
2. If the team cannot lift the crate, propose a modification to the pulley system that would allow them to lift it, assuming the rope strength remains the same.
Read the worked answer
They need to reduce the required effort to ≤1200 N. Currently effort = 1225 N. They need a mechanical advantage such that effort ≤1200 N. Required ideal effort = actual effort * efficiency = 1200 * 0.8 = 960 N. So required MA = load / ideal effort = 1960 / 960 ≈ 2.04. So they need a mechanical advantage of at least 3 (since MA is typically integer). With MA=3, ideal effort = 1960/3 ≈ 653.3 N, actual effort = 653.3/0.8 ≈ 816.7 N, which is within 1200 N. Also, tension would be 816.7 N, well below rope strength. So adding an additional pulley to increase MA to 3 would work.