Question

A wheel of radius $2\,\text{cm$ is at rest on the horizontal surface. A point $P$ on the circumference of the wheel is in contact with the horizontal surface. When the wheel rolls without slipping on the surface, the displacement of point $P$ after half rotation of wheel is}

• A. $2(\pi^2 + 2)^{1/2}$ cm
• B. $(\pi^2 + 2)^{1/2}$ cm
• C. $(\pi^2 + 4)^{1/2}$ cm
• D. $2(\pi^2 + 4)^{1/2}$ cm

Hint:

In rolling motion, track horizontal and vertical displacements separately before combining them.

Answer 1

The Correct Option is D

Step 1: Displacement of centre of wheel.
For half rotation, the centre moves a horizontal distance equal to half the circumference:
\[ x = \pi R = \pi \times 2 = 2\pi \,\text{cm} \]

Step 2: Vertical displacement of point $P$.
Initially, point $P$ is at the bottom. After half rotation, it comes to the top.
Vertical displacement:
\[ y = 2R = 4 \,\text{cm} \]

Step 3: Resultant displacement.
\[ \text{Displacement} = \sqrt{x^2 + y^2} = \sqrt{(2\pi)^2 + 4^2} \] \[ = \sqrt{4\pi^2 + 16} = 2\sqrt{\pi^2 + 4} \]

Step 4: Conclusion.
The displacement of point $P$ is $2(\pi^2 + 4)^{1/2}$ cm.