Question

List I describe four systems, each with two particles, A and B, in relative motion, as shown in the figures. List II gives the possible magnitude of their relative velocities (in m s^–1) at time \(t = \frac{\pi}{3} s.\)
 

List-I		List-II
I	A and B are moving on a horizontal circle of radius 1 m with uniform angular speed ω = 1 rad s–1. The initial angular positions of A and B at time t = 0 are θ = 0 and θ = \(\frac{\pi}{2}\), respectively.	P	\(\frac{\sqrt{3}+1}{2}\)
II	Projectiles A and B are fired (in the same vertical plane) at t = 0 and t = 0.1 s respectively,with the same speed \(v=\frac{5\pi}{\sqrt{2}}\)m s–1 and at 45° from the horizontal plane. The initial separation between A and B is large enough so that they do not collide,(g =10 m s -2 ).	Q	\(\frac{\sqrt{3}-1}{\sqrt{2}}\)
III	Two harmonic oscillators A and B moving in the x direction according to \(x_A = x_0 sin\frac{t}{t_0}\) and \(x_B=x_0 sin(\frac{t}{t_0}+\frac{\pi}{2})\) respectively, starting from t = 0. Take x0 = 1 m, t0 = 1 s.	R	\(\sqrt{10}\)
IV	Particle A is rotating in a horizontal circular path of radius 1 m on the xy plane, with constant angular speed ω = 1 rad s–1. Particle B is moving up at a constant speed 3 ms–1 in the vertical direction as shown in the figure. (Ignore gravity)	S	\(\sqrt{2}\)
		T	\(\sqrt{25\pi^{2}+1}\)

Which one of the following options is correct?

• A. I → R, II → T, III → P, IV → S
• B. I → S, II → P, III → Q, IV → R
• C. I → S, II → T, III → P, IV → R
• D. I → T, II → P, III → R, IV → S

Answer 1

The Correct Option is C

Given Information:

• System I involves particles A and B moving in horizontal circles with a radius of 1 meter at uniform angular speeds.
• System II involves projectiles A and B fired at the same speed, in the same direction, but with different angles.
• System III involves two harmonic oscillators, A and B, moving in the x direction with different angular frequencies.
• System IV involves particle A rotating in a horizontal circular path at constant speed.

System I: Particles A and B Moving in Horizontal Circles

The initial angular position of A and B at time \( t = 0 \) is \( \theta_A = 0^\circ \) and \( \theta_B = 0^\circ \), respectively.
The relative velocity between the two particles can be determined by considering their tangential velocities. Given the same radius and angular speeds, the relative velocity would be the difference in their tangential speeds.
Using the relationship \( v = r \omega \), the relative velocity magnitude is \( \sqrt{2} \), matching with option R.

System II: Projectiles A and B Fired at Same Speed, Same Angle 

Since both projectiles are fired at the same speed and at the same angle, their relative velocity will be constant throughout the motion, which gives a magnitude of \( \sqrt{2} \).
This corresponds to option Q.

System III: Two Harmonic Oscillators A and B Moving in the x Direction

- For two harmonic oscillators with the same angular frequency \( \omega \) but different initial phases, their relative motion will result in a sinusoidal variation in their displacement, giving a maximum relative velocity of \( \sqrt{2} \).
This matches with option S.

System IV: Particle A Rotating in a Horizontal Circular Path

Particle A is moving in a circular path with a constant speed \( v = 3 \, \text{m/s} \), and its relative velocity with respect to a reference point is calculated using angular velocity and position.
The relative velocity magnitude in this case is \( \sqrt{3} \), which corresponds to option K.

Answer:

The correct match of options is:

• System I: R
• System II: Q
• System III: S
• System IV: K

The correct answer is Option C: (I) - R, (II) - Q, (III) - S, (IV) - K.