Work — Question 8

Question 8 – Solution

1. Separate the bucket and rope contributions.

The bucket has constant weight, while different pieces of rope rise different distances:$$W_{\text{total}}=W_{\text{bucket}}+W_{\text{rope}}.$$

2. Calculate the bucket work.

The $40$ N bucket rises the full $12$ m:$$W_{\text{bucket}}=40(12)=480\ \mathrm{J}.$$

3. Define a consistent coordinate for the rope.

Let $x$ be the initial depth below the top of the well, with $0\le x\le12$. A segment of length $dx$ weighs $3\,dx$ newtons and rises $x$ meters.

4. Integrate the rope work.

$$W_{\text{rope}}=\int_0^{12}3x\,dx=\left[\frac32x^2\right]_0^{12}=\frac32(144)=216\ \mathrm{J}.$$

5. Add and check the total.

$$W_{\text{total}}=480+216=\boxed{696\ \mathrm{J}}.$$As a check, the rope weighs $3(12)=36$ N and its center of mass rises $6$ m, giving $36(6)=216$ J.