Question

The radius of circle, the period of revolution, initial position and sense of revolution are indicated in the fig.

y- projection of the radius vector of rotating particle P is:

• A. \(y(t)=-3 cos2\pi t\), where \(y\) in \(m\)
• B. \(y(t)=4 sin(\frac{\pi t}{2})\), where \(y\) in \(m\)
• C. \(y(t)=3cos(\frac{3\pi t}{2})\), where \(y\) in \(m\)
• D. \(y(t)=3cos(\frac{\pi t}{2})\), where \(y\) in \(m\)

Answer 1

The Correct Option is D

To solve the problem, we need to find the equation for the y-projection of the radius vector of the rotating particle \( P \) on the circle. Let's follow the steps:

1. The radius of the circle is given as \( 3 \, \text{m} \).
2. The period of revolution \( T \) is \( 4 \, \text{s} \). This means the particle makes one complete revolution in \( 4 \) seconds.
3. The initial position of the particle at \( t = 0 \) is along the positive y-axis.
4. The angular velocity \( \omega \) can be calculated using the formula: \(\omega = \frac{2\pi}{T}\)
   Substituting the period: \(\omega = \frac{2\pi}{4} = \frac{\pi}{2}\) rad/s.
5. The angle \(\theta\) at time \( t \) is given by \(\omega t = \frac{\pi t}{2}\).
6. Since the particle starts at the positive y-axis, the y-projection of the position vector at any time \( t \) is given by: \(y(t) = 3 \cos(\theta)\). Substituting \(\theta\): \(y(t) = 3 \cos\left(\frac{\pi t}{2}\right)\).

Thus, the y-projection of the radius vector of the rotating particle \( P \) is given by the equation \(y(t) = 3 \cos\left(\frac{\pi t}{2}\right)\), where \( y \) is in meters.

This matches the correct option:

• \(y(t)=3\cos\left(\frac{\pi t}{2}\right)\), where \(y\) in \(m\)