Class 11 > Unit # 04: Rotational & Circular Motion > Centripetal Force & Acceleration

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Unit No. 4 Rotational and Circular Motion - Centripetal Acceleration & Centripetal Force

Define Centripetal Force and Centripetal Acceleration, and Derive Expression for Centripetal Acceleration.

Centripetal Acceleration

"The acceleration produced in a body moving along a circular path due to the continuous change in the direction of its velocity is called centripetal acceleration. It is always directed towards the center of the circle or curvature."

Formula

$a_c = \frac{v^2}{r}$

Figure: Geometric representation of position and velocity vectors in uniform circular motion.

Derivation of Centripetal Acceleration

Let us consider a particle of mass $m$ moving with a uniform speed $v$ along a circular path of radius $r$. Suppose its velocity vector at point $P_1$ is $\vec{v}_1$ at time $t_1$, and its velocity vector at point $P_2$ is $\vec{v}_2$ at time $t_2$.

In uniform circular motion, the magnitude of velocity remains constant ($v_1 = v_2 = v$), but its direction changes continuously. This change in velocity vector ($\Delta v$) is entirely due to the change in direction. The angle between the two velocity vectors $\vec{v}_1$ and $\vec{v}_2$ is equal to the angle $\Delta\theta$ subtended by the radial lines at the center.

From geometry, the triangle formed by the position vectors (radius $r$ and arc length $\Delta S$) and the triangle formed by the velocity vectors ($\vec{v}_1$, $\vec{v}_2$, and $\Delta v$) are similar (congruent triangles ratio):

$\frac{\Delta v}{v} \approx \frac{\Delta S}{r}$

Rearranging the equation for $\Delta v$:

$\Delta v = \frac{v}{r} \Delta S$

Dividing both sides by the time interval $\Delta t$:

$\frac{\Delta v}{\Delta t} = \frac{v}{r} \frac{\Delta S}{\Delta t}$

Taking the limit as $\Delta t \to 0$, the left side becomes instantaneous acceleration $a_c$, and $\frac{\Delta S}{\Delta t}$ becomes linear speed $v$:

$a_c = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{v}{r} \left( \lim_{\Delta t \to 0} \frac{\Delta S}{\Delta t} \right)$

Substituting $\lim_{\Delta t \to 0} \frac{\Delta S}{\Delta t} = v$ into the equation:

$a_c = \frac{v}{r} (v)$

$a_c = \frac{v^2}{r} \quad \text{(Proved)}$

Centripetal Force

"The force that compels an object to move along a curved path and points toward the center of curvature is called centripetal force."

Mathematical Expression

According to Newton's Second Law of Motion ($F = ma$), substituting centripetal acceleration ($a_c = \frac{v^2}{r}$):

$F_c = m \cdot a_c$

$F_c = \frac{mv^2}{r}$

Where:
• $F_c$ = Centripetal Force
• $m$ = Mass of object
• $v$ = Velocity of object
• $r$ = Radius of the curved path

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