What is Centripetal Force?
Centripetal force is the net force acting on an object moving in a curved path, directed toward the center of curvature. It is responsible for changing the direction of the velocity vector without changing the speed of the object.
Deriving Centripetal Acceleration ($a = v^2/r$)
Consider a particle moving along a circular path of radius $r$ with constant speed $v$.
- Define Positions: Let the particle move from point $A$ to $B$ in a small time interval $\Delta t$. The angular displacement is $\Delta \theta$.
- Velocity Vectors: At point $A$, the velocity is $\vec{v_A}$. At point $B$, it is $\vec{v_B}$. Since the speed is constant, $|\vec{v_A}| = |\vec{v_B}| = v$.
- Change in Velocity: The change in velocity is $\Delta \vec{v} = \vec{v_B} - \vec{v_A}$.
- Geometric Similarity: The triangle formed by the position vectors (radius $r$) is similar to the triangle formed by the velocity vectors (speed $v$) because both have the same angle $\Delta \theta$ between sides.
- Ratio Equality: From similarity: $\frac{|\Delta \vec{v}|}{v} = \frac{\text{arc } AB}{r} = \frac{v \Delta t}{r}$
- Limit Approach: As $\Delta t \to 0$, the ratio $\frac{\Delta \vec{v}}{\Delta t}$ becomes the instantaneous acceleration $a$. Therefore: $a = \frac{v^2}{r}$
Direction of Acceleration
The acceleration vector always points toward the center of the circle. This is because $\Delta \vec{v}$ points inward as $\Delta t$ approaches zero, consistent with the force maintaining the circular orbit.