Question

A disc is rolling without slipping on a surface. The radius of the disc is R. At t = 0, the top most point on the disc is A as shown in figure. When the disc completes half of its rotation, the displacement of point A from its initial position is

• A. \(2R\sqrt{1+4π^2}\)
• B. 2R
• C. \(R\sqrt{π^2+1}\)
• D. \(R\sqrt{π^2+4}\)

Hint:

For rolling motion without slipping, calculate displacement considering both horizontal and vertical components. The horizontal motion is determined by the arc length, and the vertical motion depends on the rolling geometry.

Answer 1

The Correct Option is D

When the disc rolls without slipping, the topmost point \( A \) moves along a cycloidal path. The total displacement is calculated as follows:

Step 1: Distance moved by the center of the disc.
The center of the disc moves a horizontal distance equal to the arc length subtended by half the circumference:
\[ \text{Horizontal displacement} = \pi R. \]

Step 2: Vertical displacement of point \( A \).
Point \( A \) starts at the top of the disc and ends at the bottom after half a rotation, so the vertical displacement is:
\[ \text{Vertical displacement} = 2R. \]

Step 3: Resultant displacement.
The resultant displacement is given by:
\[ \text{Displacement} = \sqrt{(\pi R)^2 + (2R)^2}. \]

Substitute the values:
\[ \text{Displacement} = \sqrt{\pi^2 R^2 + 4R^2}. \]

Factor \( R^2 \):
\[ \text{Displacement} = R \sqrt{\pi^2 + 4}. \]