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A particle's position vector is given by \(\vec r=\cos(\omega t)\,\hat x+\sin(\omega t)\,\hat y\), where \(\omega\) is a constant. Which of the following correctly describes the motion?

1. Velocity is perpendicular to \(\vec r\) and acceleration points away from the origin.
2. Both velocity and acceleration are perpendicular to \(\vec r\).
3. Both velocity and acceleration are parallel to \(\vec r\).
4. Velocity is perpendicular to \(\vec r\) and acceleration points toward the origin.

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

Solution

The given position vector has constant magnitude 1, so the particle moves on a circle. Differentiating gives velocity perpendicular to \(\vec r\), and the second derivative gives acceleration opposite to \(\vec r\), i.e. toward the origin.

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