A body of mass m is pushed slowly and uniformly up a curved…

A body of mass $m$ is pushed slowly and uniformly up a curved surface by a force $\vec F$ that is always tangent to the path. Find the work of $\vec F$ between points 1 and 2, which differ in height by $h$ and are $l$ apart horizontally. The coefficient of friction is $\mu$.

Explanation of Method

Work to push a body up a curved profile.

Step 1: A short piece of the path

On a piece $ds$ inclined at $\theta$, uniform (slow) motion means $F = mg\sin\theta + \mu mg\cos\theta$, so

\[ dW = F\,ds = mg\,\sin\theta\,ds + \mu mg\,\cos\theta\,ds, \]

\[ dW = mg\,dy + \mu mg\,dx. \]

Step 2: Whole path

Adding the pieces, $\sum dy = h$ and $\sum dx = l$, whatever the shape: $W = mgh + \mu mg\,l$.

$1$ $2$ $l$ $h$ $\vec F$ $\mu N$ $W = mgh + \mu mg\,l$

On every short piece of the path, friction $\mu mg\cos\theta\,ds = \mu mg\,dx$ depends only on the horizontal advance, so the shape of the profile does not matter.

Result

\[\boxed{\,W = mgh + \mu mg\,l\,}\]

Remark

Check. The original result is correct, but its derivation treats the path as a single straight incline. The step above shows why it holds for any profile. It assumes the motion is slow enough that the centripetal term $mv^2/\rho$ in the normal force is negligible on the curved parts.