What is centripetal acceleration? Derive an expression for centripetal acceleration of a particle in uniform circular motion along a plane. What will be the direction of the velocity and acceleration at any instant? (1+3+1=5)

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Accepted Answer

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> [!TLDR]
> Centripetal acceleration is the radially-inward acceleration that continuously changes the *direction* (not the speed) of a particle in uniform circular motion; it is derived here from the time-derivative of the position vector, and its direction is shown to be always perpendicular to the (tangential) velocity.
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> **(i) What is centripetal acceleration?** In uniform circular motion, a particle moves with constant speed $v$ along a circle of radius $r$, but its velocity direction keeps changing. This continuous change of direction requires an acceleration, directed at every instant towards the centre of the circle — this is called centripetal ('centre-seeking') acceleration.
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> **(ii) Derivation of the expression for centripetal acceleration:**
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> Let the particle move on a circle of radius $r$ in the xy-plane, with angular speed $\omega$ (constant, for uniform circular motion). At time $t$, its position vector is:
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> $\vec r(t) = r\cos(\omega t)\,\hat i + r\sin(\omega t)\,\hat j$
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> **Velocity:**
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> $\vec v(t) = \dfrac{d\vec r}{dt} = -r\omega\sin(\omega t)\,\hat i + r\omega\cos(\omega t)\,\hat j$
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> Its magnitude is $|\vec v| = r\omega\sqrt{\sin^2(\omega t)+\cos^2(\omega t)} = r\omega = v$ (constant, as expected for uniform circular motion), and $v=r\omega$.
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> **Acceleration:**
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> $\vec a(t) = \dfrac{d\vec v}{dt} = -r\omega^2\cos(\omega t)\,\hat i - r\omega^2\sin(\omega t)\,\hat j = -\omega^2\,\vec r(t)$
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> So the acceleration vector is directed exactly opposite to the position vector $\vec r$ (which points from the centre to the particle) — i.e. $\vec a$ points from the particle towards the centre of the circle.
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> Its magnitude: $|\vec a| = \omega^2 r$. Using $v=r\omega \Rightarrow \omega = v/r$:
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> $a_c = \omega^2r = \left(\dfrac{v}{r}\right)^2 r = \dfrac{v^2}{r}$
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Q.What is centripetal acceleration? Derive an expression for centripetal acceleration of a particle in uniform circular motion along a plane. What will be the direction of the velocity and acceleration at any instant? (1+3+1=5)

Concept understanding — Uniform Circular Motion

Uniform Circular Motion

1. Angular Quantities

2. Centripetal Acceleration

3. Directions

4. Points on a Rotating Body

5. Connected Systems

6. Non-Uniform Circular Motion

Q.What is centripetal acceleration? Derive an expression for centripetal acceleration of a particle in uniform circular motion along a plane. What will be the direction of the velocity and acceleration at any instant? (1+3+1=5)

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