newtonian mechanics - Swinging of a bob

Suppose a variable force is acting on the bob and making it swing as shown in the figure then why will kinetic energy be conserved here even though there is a variable force acting on the bob?

Answer

This is a straight-forward application of the mechanical work-energy theorem: the net work done by all forces acting on a body as it moves from position $i$ to position $f$ is equal to the change in the kinetic energy of the body, $\Delta K = K_f-K_i$ or

$$\sum_{\mathrm{all~forces}}W = \Delta K.$$ It doesn't matter what the nature and origins of the forces are. All that matters is whether they do work on the object (mechanically) or not. They may be conservative or non-conservative, time-dependent or constant.

Kinetic energy is not conserved, and the calculation of kinetic energy is frame-dependent, so this principle/theorem must be applied wisely.

The work energy theorem arises out of the conservation of energy when dealing with mechanical processes (and may carefully be extended to other processes, but we won't deal with that here):

$$E_f=E_i+W_{\mathrm{non-conservative}}$$ where $E=K+U$, and $U$ is the system potential energy associated with some conservative force. As part of accounting for energy changes in a system, we either count the change in a potential energy or the work done by that conservative force, but not both. We also define changes in potential energy as $$\Delta U_n = -W_n,$$ where the $n$ subscript identifies a particular conservative force.

Replacing the $E_f$ and $E_i$ by $K+U$ we get the following:

$$K_f + \sum_n U_{nf}= K_i + \sum_n U_{ni} +W_{\mathrm{non-conservative}}$$ $$K_f - K_i = \sum_n\left(-U_{nf}+U_{ni} \right)+W_{\mathrm{non-conservative}}$$ $$K_f - K_i = \sum_n(W_n)+W_{\mathrm{non-conservative}}$$ $$\Delta K = W_{\mathrm{all}}$$

If the kinetic energies at two locations along a path are the same, then $\Delta K=0$ and the net work done by all external forces acting on the object while traveling between those two points will also be zero (not including thermodynamic effects). The variety of changes to velocities and speeds between those two points are irrelevant.