Work — Question 4

Question 4 – Solution

1. Choose a coordinate for each cable segment.

Let $x$ measure the initial distance downward from the top of the building. The cable occupies $0\le x\le6$ meters.

2. Find the weight of a small segment.

The given quantity is weight per unit length. A segment of length $dx$ weighs$$dF=(4\ \mathrm{N/m})\,dx.$$No additional factor of gravity is needed.

3. Find its lifting distance and work.

The segment initially at depth $x$ must rise $x$ meters. Thus$$dW=(\text{weight})(\text{distance})=4x\,dx.$$

4. Add the work for all segments.

$$W=\int_0^6 4x\,dx=\left[2x^2\right]_0^6=2(36)-0=72.$$

5. State and check the result.

$$\boxed{W=72\ \mathrm{J}}.$$The cable weighs $4(6)=24$ N, and its center of mass rises $3$ m. This gives the same work: $24(3)=72$ J.