Question

A 4-meter chain with linear mass density (x) = 2x(7−x) kg/m lies on the ground. Calculate the work required to lift the chain from its front end so that its bottom is 3 m above the ground.

Answer 1

The work to lift a chain with a non-uniform linear mass density that varies with position is calculated by integrating the product of each differential mass element's weight and the height it is lifted. This calculation requires an understanding of variable mass systems and the concept of work in physics.

Step-by-step explanation:

The question asks for the work required to lift a non-uniform chain so that its bottom is 3 meters above the ground. The linear mass density of the chain varies with position according to μ(x) = 2x(7−x), where x is the position along the chain. To calculate the work done, we integrate the weight of each differential element of the chain over the height to which it is raised. As we lift the chain, elements closer to the front end are lifted higher than those nearer to the bottom end.

We consider a differential element of the chain, dm, at position x with length dx. This element has a mass of μ(x)dx = 2x(7−x)dx. The work dW done to lift this element is the product of its weight dmg and the height h it is lifted, where g is the acceleration due to gravity and h is (3 meters - x), because the bottom is lifted to 3 meters. Thus, dW = dmg h = 2x(7−x)g(3 - x)dx. Integrating this expression from x = 0 to x = 4 meters gives the total work done.

To solve this problem, we would perform the integration and find the numerical solution. Note that the units of work are joules (J).