Question

The position vector of a particle is \(\overset{⃗}{r} = acosωt\hat{i}+asinωt\hat{j}\). Calculate the angle between its position vector and the velocity vector.
(\(cos0 = 1,cos90 = 0,cos60 = 0.5,sin30 = 0.5,sin0 = 0\))

• A. \(30^{\circ}\)
• B. \(60^{\circ}\)
• C. \(90^{\circ}\)
• D. \(0^{\circ}\)

Hint:

Check the dot product of $\vec r$ and $\vec v$.

Answer 1

The Correct Option is C

Step 1: Find velocity
\(\vec v=\frac{d\vec r}{dt}=-a\omega\sin\omega t\,\hat i+a\omega\cos\omega t\,\hat j\).

Step 2: Dot product
\(\vec r\cdot\vec v=-a^2\omega\sin\omega t\cos\omega t+a^2\omega\sin\omega t\cos\omega t=0\).

Step 3: Angle
A zero dot product means the vectors are perpendicular, so the angle is \(90^{\circ}\). Option (C).

Final Answer:
The angle is \(90^{\circ}\), option (C). \[ \boxed{\text{(C) }90^{\circ}} \]