A person exerts a constant force F on a particle of mass m and...

For a particle moving in a circle of radius $r$ with uniform speed $v$, the net force acting on it must have a constant magnitude of $\dfrac{mv^2}{r}$ and be directed towards the centre of the circular path.

Let this net force be $\vec{F}_{net}$. Since the position of the particle continuously changes, the direction of $\vec{F}_{net}$ also continuously changes. Thus, $\vec{F}_{net}$ is a variable force.

The person exerts a constant force $\vec{F}$. For the particle to experience the required net force $\vec{F}_{net}$, there must be additional forces acting on it. Let the resultant of these additional forces be $\vec{F}_{add}$.

We can write $\vec{F}_{net} = \vec{F} + \vec{F}_{add}$, which gives $\vec{F}_{add} = \vec{F}_{net} - \vec{F}$.

The magnitude of the additional force is given by $|\vec{F}_{add}|^2 = |\vec{F}_{net}|^2 + |\vec{F}|^2 - 2|\vec{F}_{net}||\vec{F}|\cos\theta$, where $\theta$ is the angle between $\vec{F}_{net}$ and $\vec{F}$.

As the particle moves along the circle, the direction of $\vec{F}_{net}$ rotates while $\vec{F}$ remains constant. Therefore, the angle $\theta$ continuously changes, causing the magnitude of $\vec{F}_{add}$ to vary. Its direction must also vary to ensure that the vector sum with the constant force $\vec{F}$ always points towards the centre.