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A particle moves so that its position vector is given by \(\vec r = \hat i\cos \omega t + \hat j\sin \omega t\). Which of the following is true?

1. Velocity is perpendicular to \(\vec r\) and acceleration is directed away from the origin.
2. Velocity and acceleration both are perpendicular to \(\vec r\).
3. Velocity and acceleration both are parallel to \(\vec r\).
4. Velocity is perpendicular to \(\vec r\) and acceleration is directed towards the origin.

Correct answer: Velocity is perpendicular to \(\vec r\) and acceleration is directed towards the origin.

Solution

For \(\vec r = \hat i\cos\omega t + \hat j\sin\omega t\), velocity is \(\vec v = -\omega\hat i\sin\omega t + \omega\hat j\cos\omega t\), which is perpendicular to \(\vec r\). Acceleration is \(\vec a = -\omega^2\hat i\cos\omega t - \omega^2\hat j\sin\omega t = -\omega^2\vec r\), so it points towards the origin.

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