Lifting problem A 4-kg mass is attached to the bottom of a 5-m, 15-kg chain. If the chain hangs from a platform, how much work is required to pull the chain and the mass onto the platform?

Verified step by step guidance

1. Identify the physical setup: a 5-meter chain with mass 15 kg hangs from a platform, and a 4-kg mass is attached at the bottom. We want to find the work done to pull both the chain and the mass up onto the platform.
2. Calculate the weight per unit length of the chain. Since the chain has mass 15 kg and length 5 m, the linear mass density is \(\lambda = \frac{15}{5} = 3\) kg/m. The weight per unit length is then \(w = \lambda g = 3g\) N/m, where \(g\) is the acceleration due to gravity.
3. Set up the integral for the work done to lift the chain. Consider a small segment of the chain at a distance \(x\) meters from the bottom (where \(x\) ranges from 0 to 5). The length of chain to be lifted from this segment is \(x\) meters, so the work to lift this segment is \(dW = (\text{weight of segment}) \times (\text{distance lifted})\).
4. Express the work to lift the chain segment as \(dW = (w \, dx) \times x = 3g x \, dx\). Integrate this expression from \(x=0\) to \(x=5\) to find the total work to lift the entire chain.
5. Calculate the work to lift the 4-kg mass from the bottom to the platform, which is a distance of 5 meters. The work is \(W_{mass} = mg \times 5 = 4g \times 5\). Add this to the work done lifting the chain to get the total work required.

Key Concepts

Work Done by a Variable Force

Work is the integral of force over distance. When lifting a chain, the force varies because the weight of the portion lifted changes as more chain is pulled up. Calculating work requires integrating the weight of the chain segment being lifted over the height it is moved.

Weight and Mass Relationship

Weight is the force due to gravity acting on a mass, calculated as mass times gravitational acceleration (W = mg). For the chain and the mass, their weights must be considered separately and combined to find the total force during lifting.

Setting up the Integral for Work

To find the total work, set up an integral that sums the incremental work done lifting each small segment of the chain plus the constant work to lift the mass. This involves expressing the force as a function of the height lifted and integrating over the chain length.