Consider a disc rotating in the horizontal plane with a constant angular speed to about its centre O. The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the figure. When the disc is in the orientation as shown, two pebbles P and Q are simultaneously projected at an angle towards R The velocity of projection is in the y-z plane and is same for both pebbles with respect to the disc. Assume that (i) they land back on the disc before the disc has completed 1/8 rotation, (ii) their range is less than half the disc radius, and (iii) \(\omega\) remains constant throughout. Then
- P lands in the shaded region and Q in the un shaded region
- P lands in the unshaded region and Q in the shaded region
- both P and Q land in the unshaded region
- both P and Q land in the shaded region
The Correct Option is D
Solution and Explanation
Since the disc completes \(\frac{1}{8}\) of a rotation, the time for \(\frac{1}{8}\) rotation is \(\frac{T}{8}\), where T is the period of the disc.
The period T is given by \(T=\frac{2\pi}{\omega}\)
Therefore, the time for 1/8 rotation is \(t = \frac{T}{8} = \frac{2\pi}{8\omega} = \frac{\pi}{4\omega}\)
X- coordinate of P = ωRt
\(= \frac{πR}{4} \gt Rcos45\degree\)
Therefore, P and Q lands in the unshaded region.
So. the correct option is (D): both P and Q land in the shaded region
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Concepts Used:
System of Particles and Rotational Motion
- The system of particles refers to the extended body which is considered a rigid body most of the time for simple or easy understanding. A rigid body is a body with a perfectly definite and unchangeable shape.
- The distance between the pair of particles in such a body does not replace or alter. Rotational motion can be described as the motion of a rigid body originates in such a manner that all of its particles move in a circle about an axis with a common angular velocity.
- The few common examples of rotational motion are the motion of the blade of a windmill and periodic motion.