If a particle is moving on an elliptical path given by r⃗=b cosωt 1̂+a sinωt^n, then its radial

step 1 In this question it is told that a particle is moving on an elliptical path and we have given it

If a particle is moving on an elliptical path given by r⃗=b...

If a particle is moving on an elliptical path given by $\vec{r}=b \cos \omega t \hat{1}+a \sin \omega t^{n}$, then its radial acceleration along $\vec{r}$ is : (a) $\omega \overrightarrow{\mathrm{r}}$ (b) $\omega^{2} \vec{r}$ (c) $-\omega^{2} \overrightarrow{\mathbf{r}}$ (d) none of these

If a particle is moving on an elliptical path given by $\vec{r}=b \cos \omega t \hat{1}+a \sin \omega t^{n}$, then its radial acceleration along
$\vec{r}$ is :
(a) $\omega \overrightarrow{\mathrm{r}}$
(b) $\omega^{2} \vec{r}$
(c) $-\omega^{2} \overrightarrow{\mathbf{r}}$
(d) none of these

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Key Concepts

Radial (Centripetal) Acceleration

Radial or centripetal acceleration refers to the component of the acceleration that is directed along the position vector (or towards the center of curvature of the path). It quantifies the rate at which the speed of the particle's direction is changing and, in cases like uniform circular motion or certain elliptical motions, it can be directly proportional to the negative of the position vector, thereby pointing inwards.

Parametric Equations of Motion

This concept involves describing the position of a moving particle using functions of a parameter, typically time. In many cases including elliptical or circular paths, the coordinates of the particle are given as separate functions (such as sine and cosine functions) that together specify the entire trajectory in the plane or space.

Vector Differentiation

Vector differentiation is the process of taking the derivative of vector-valued functions with respect to time. In kinematics, differentiating the position vector once gives the velocity vector and differentiating a second time gives the acceleration vector. This technique is crucial for analyzing the motion of objects along trajectories defined by parametric equations.

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