Work done my tangential force in non uniform circular motion

Question

Why is the work done by the tangential force in non uniform circular motion considered to be non zero? After one complete revolution the net displacement is 0. According to my understanding if the net displacement is 0 the work done by the tangential force is also 0. I am unable to understand were am I going wrong

Answer 1

Because the equation you're using $W=\mathbf {\vec F}\cdot \Delta {\bf \vec d}$ is only applicable when the force $\bf \vec F$ is constant.

The proper work definition is obtained as follows: for a particle being acted upon by some force $\bf \vec F$, we can take a tiny displacement $\delta {\bf \vec r}$, and take its dot product with the force to get the tiny amount of work $\delta W$ done over that tiny displacement $\delta {\bf \vec r}$. In other words, the tiny amount of work is $\delta W={\bf \vec F}\cdot \delta {\bf \vec r}$. Then, the total work is $\displaystyle W=\sum {\bf \vec F}\cdot \delta {\bf \vec r}$, which is expressed as an integral as follows: $$W_{\rm by \ \bf \vec F}=\int_C{\bf \vec F}\cdot \mathrm d{\bf \vec r}$$ where $C:{\bf \vec r}(t)$ is the path the particle takes.

For a particle undergoing circular motion, the tangential force accelerating it keeps changing direction (since it's always tangent to the path), and thus is not constant.

If the tangential force is constant in magnitude (but not in direction), then the work done will be $W=Fs$ where $s$ is the distance over which the force acted.

The wiki article is quite good: https://en.wikipedia.org/wiki/Work_(physics).

Answer 2

The work done by a conservative force is 0 in the case you are describing . The forces which are non conservative in nature (for example friction ) can have non zero work done in a circular loop.