Question

A circular portion of radius $R_2$ has been removed from one edge of a circular disc of radius $R_1$. The correct expression for the centre of mass for the remaining portion of the disc is -

• A. $-\frac{R_2^2}{R_1+R_2}$
• B. $-\frac{R_1^2}{R_1+R_2}$
• C. $\frac{R_2^2}{R_1+R_2}$
• D. $-\frac{R_1^2}{R_1+R_2}$

Answer 1

Answer: (A)

$-\frac{R_2^2}{R_1+R_2}$

Answer 2

To find the center of mass of the remaining portion of the disc, we use the principle of superposition. We consider the original complete disc as a system and the removed portion as having negative mass.

1. Define Coordinate System and Masses:

   Let the original circular disc of radius $R_1$ be centered at the origin $(0,0)$. Let $\sigma$ be the uniform surface mass density of the disc. The mass of the original disc is $M_1 = \sigma \cdot (\text{Area}_1) = \sigma \pi R_1^2$. The center of mass of the original disc is $X_1 = 0$.

2. Locate the Removed Portion:

   A circular portion of radius $R_2$ is removed from one edge. The standard interpretation of this phrase in physics problems is that the removed disc is tangent to the edge of the original disc. Let's assume the removal occurs along the positive x-axis. The rightmost point of the original disc is $(R_1, 0)$. For the removed disc to be tangent to this point and entirely contained within the original disc, its center must be at a distance $R_2$ to the left of this point. So, the x-coordinate of the center of the removed disc is $X_2 = R_1 - R_2$. The mass of the removed disc is $M_2 = \sigma \cdot (\text{Area}_2) = \sigma \pi R_2^2$.

3. Calculate the Center of Mass of the Remaining Portion:

   The center of mass of the remaining portion ($X_{CM}$) is given by the formula:

   $X_{CM} = \frac{M_1 X_1 - M_2 X_2}{M_1 - M_2}$

   Substitute the values:

   $X_{CM} = \frac{(\sigma \pi R_1^2)(0) - (\sigma \pi R_2^2)(R_1 - R_2)}{\sigma \pi R_1^2 - \sigma \pi R_2^2}$

   The term $(\sigma \pi)$ can be factored out from both the numerator and the denominator and cancelled:

   $X_{CM} = \frac{- R_2^2 (R_1 - R_2)}{R_1^2 - R_2^2}$

   Recognize the denominator as a difference of squares: $R_1^2 - R_2^2 = (R_1 - R_2)(R_1 + R_2)$.

   $X_{CM} = \frac{- R_2^2 (R_1 - R_2)}{(R_1 - R_2)(R_1 + R_2)}$

   Assuming $R_1 \neq R_2$ (otherwise, the entire disc is removed, or the problem is ill-defined), we can cancel the $(R_1 - R_2)$ term:

   $X_{CM} = -\frac{R_2^2}{R_1 + R_2}$

   The negative sign indicates that the center of mass shifts in the direction opposite to where the mass was removed (i.e., to the left of the original center).