Q.Give one example each of central force and non-central force.

The key idea here is distinguishing between forces based on their direction and dependence on position.

A central force is a force whose direction is always along the line joining the particle to a fixed point (the center of force), and whose magnitude depends only on the distance from that point.

For example, the gravitational force between two point masses is always directed along the line connecting their centers and its magnitude depends only on the distance between them.

A non-central force is a force that does not satisfy these conditions. Its direction may not be along the position vector, or its magnitude may depend on factors other than just the distance from a fixed point. …

The distinction between central and non-central forces is fundamental in physics, particularly in classical mechanics, because it dictates whether certain conservation laws, like the conservation of angular momentum, apply.

A force is defined as a central force if it satisfies two conditions:

1. It is always directed along the line joining the centers of the two interacting bodies.
2. Its magnitude depends only on the distance between these two bodies. Mathematically, a central force $\vec{F}$ acting on a particle at position $\vec{r}$ (relative to the force center) can be written as $\vec{F}(\vec{r}) = f(r) \hat{r}$, where $r = |\vec{r}|$ is the magnitude of the position vector, $f(r)$ is a scalar function of the distance $r$ only, and $\hat{r} = \vec{r}/r$ is the unit vector in the radial direction.

A force is a non-central force if it does not satisfy one or both of these conditions. That is, it might not act along the line joining the centers, or its magnitude might depend on factors other than just the distance, such as the direction of motion, velocity, or the orientation of the interacting bodies.

Let's look at examples for each.

Example of a Central Force

1. Gravitational Force: Consider two point masses, $m_1$ and $m_2$, separated by a distance $r$. The gravitational force exerted by $m_1$ on $m_2$ is given by Newton's Law of Universal Gravitation:

Here, $G$ is the gravitational constant, and $\hat{r}$ is the unit vector pointing from $m_1$ to $m_2$.

*   **Condition 1 (Direction):** The force $\vec{F}_g$ is directed along the line joining the centers of the two masses (the negative sign indicates it's an attractive force, pulling $m_2$ towards $m_1$).
*   **Condition 2 (Magnitude):** The magnitude of the force, $|\vec{F}_g| = G \frac{m_1 m_2}{r^2}$, depends only on the masses $m_1$, $m_2$ (which are constants) and the distance $r$ between them. It does not depend on the velocity of the masses, their orientation, or any other factor.

Since both conditions are met, the gravitational force is a classic example of a central force.

Example of a Non-Central Force

1. Magnetic Force on a Moving Charge: …

Step 1: a central force F(r)=f(r) r_hat acts along the line joining two bodies, magnitude depends only on separation. Step 2: gravity, F=-G m1 m2/r^2 r_hat, is a standard example. Step 3: a non-central force fails one/both conditions, e.g. depends on velocity or acts perpendicular to the joining line. Step 4: the magnetic force …