Angular momentum of the particle rotating with a central force is constant due to

The angular momentum of a particle is related to the torque acting on it by the rotational analogue of Newton's second law:

$$\tau = \frac{dL}{dt}$$

Here, $\tau$ is the torque and $$L$$ is the angular momentum.

For a particle moving under the influence of a central force, the line of action of the force always passes through a fixed point (the center of force). The torque of this central force about the center of force is given by:

$$\tau = \vec{r} \times \vec{F}$$

Since a central force $$\vec{F}$$ is always directed along or opposite to the position vector $$\vec{r}$$, the angle between $$\vec{r}$$ and $$\vec{F}$$ is either 0 degrees or 180 degrees. The cross product of two parallel or anti-parallel vectors is zero:

$$\tau = 0$$

Substituting $\tau = 0$ into the torque-angular momentum relation:

$$\frac{dL}{dt} = 0$$

This implies that the angular momentum $$L$$ remains constant over time (conserved). Therefore, the angular momentum remains constant due to zero torque acting on the particle.