Question

A particle moving in a circle of radius R with a uniform speed takes a time T to complete one revolution. If this particle were projected with the same speed at an angle ‘θ’ to the horizontal, the maximum height attained by it equals 4R. The angle of projection, θ, is then given by

• A. \(\theta=sin^{-1}(\frac{2gT^2}{\pi^2R})^{\frac{1}{2}}\)
• B. \(\theta=cos^{-1}(\frac{gT^2}{\pi^2R})^{\frac{1}{2}}\)
• C. \(\theta=cos^{-1}(\frac{\pi^2R}{gT^2})^{\frac{1}{2}}\)
• D. \(\theta=sin^{-1}(\frac{\pi^2R}{gT^2})^{\frac{1}{2}}\)

Answer 1

The Correct Option is A

To solve this problem, we need to relate the circular motion of a particle to its projectile motion. Let's break down the solution step-by-step:

1. The particle is moving in a circle of radius \( R \) with a uniform speed. From the motion in a circle, its speed \( v \) can be calculated using the formula for the circumference of the circle and the time period \( T \):
   • \(\text{Circumference of the circle} = 2\pi R\) 
   • \(v = \frac{2\pi R}{T}\)
2. When projected with this speed \( v \) at an angle \( \theta \), the particle's maximum height (\( H \)) in projectile motion is given by:
   • \(H = \frac{v^2 \sin^2 \theta}{2g}\)
3. According to the problem, this maximum height is equal to \( 4R \). Thus, we set up the equation:
   • \(\frac{v^2 \sin^2 \theta}{2g} = 4R\)
4. Substitute the expression for \( v \):
   • \(\frac{\left( \frac{2\pi R}{T} \right)^2 \sin^2 \theta}{2g} = 4R\)
5. Simplify the equation to solve for \( \sin^2 \theta \):
   • \(\frac{4\pi^2 R^2 \sin^2 \theta}{2gT^2} = 4R\)
   • \(\frac{2\pi^2 R \sin^2 \theta}{gT^2} = 1\)
   • \(\sin^2 \theta = \frac{gT^2}{2\pi^2 R}\)
6. Taking the square root of both sides gives:
   • \(\sin \theta = \left( \frac{gT^2}{2\pi^2 R} \right)^{1/2}\)
7. Therefore, the angle of projection \( \theta \) is:
   • \(\theta = \sin^{-1} \left( \frac{2gT^2}{\pi^2 R} \right)^{1/2}\)

Thus, the correct answer is:

\(\theta=sin^{-1}(\frac{2gT^2}{\pi^2R})^{\frac{1}{2}}\)

This option corresponds to the correct angle of projection that satisfies the given condition of maximum height being \( 4R \).