A circular disc of radius R is removed from a bigger...

A circular disc of radius $R$ is removed from a bigger circular disc of radius $2 R$ such that the circumferences of the discs coincide. The centre of mass of the new disc is $a / R$ from the centre of the bigger disc. The value of $a$ is (A) $\frac{1}{2}$ (B) $\frac{1}{6}$ (C) $\frac{1}{4}$ (D) $\frac{1}{3}$

A circular disc of radius $R$ is removed from a bigger circular disc of radius $2 R$ such that the circumferences of the discs coincide. The centre of mass of the new disc is $a / R$ from the centre of the bigger disc. The value of $a$ is
(A) $\frac{1}{2}$
(B) $\frac{1}{6}$
(C) $\frac{1}{4}$
(D) $\frac{1}{3}$

Key Concepts

Center of Mass of Composite Bodies

This concept involves finding the point at which the entire mass of a system can be considered to be concentrated. When dealing with composite bodies—structures made up of multiple parts or regions of differing mass—the overall center of mass is found by computing a weighted average of the individual centers based on their masses.

Negative Mass Concept in Center of Mass Calculations

When a portion of an object is removed, its effect on the center of mass can be accounted for by treating the missing mass as negative. This subtraction method allows one to use the same formulas for center of mass by considering removed sections as having negative mass, thereby shifting the overall balance point.

Symmetry in Mass Distribution

Symmetry simplifies center of mass calculations by ensuring that the distribution of mass is even on either side of a specific axis or point. In many problems, including those involving circular parts, symmetry allows for the reduction of a two-dimensional problem into one dimension, as the center of mass will lie along the line joining the centers of the individual symmetric components.

Geometric Properties of Circles in Mass Distribution

Understanding the area and spatial properties of circles is essential when determining mass distributions for objects with circular shapes. Since the mass of a uniform disc is proportional to its area, knowing how to compute the area from a circle’s radius helps in establishing the relative masses of different parts, which is critical for accurate center of mass computations.

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