Question

Consider a circular disc of radius 20 cm with center located at the origin. A circular hole of radius 5 cm is cut from this disc in such a way that the edge of the hole touches the edge of the disc. The distance of the center of mass of the residual or remaining disc from the origin will be:

• A. 2.0 cm
• B. 0.5 cm
• C. 1.5 cm
• D. 1.0 cm

Hint:

When calculating the center of mass of a system with a removed portion, use the formula: \[ X_{\text{com}} = \frac{m_{\text{1}} x_1 + m_{\text{2}} x_2}{m_{\text{1}} + m_{\text{2}}} \] where \( x_1 \) and \( x_2 \) are the distances of the centers of mass of the original and removed portions, and \( m_1 \) and \( m_2 \) are their respective masses.

Answer 1

The Correct Option is D

The remaining disc is formed by cutting a smaller disc from a larger disc. To find the center of mass of the remaining portion, we need to find the center of mass of the entire disc and subtract the center of mass of the cut portion.
The mass of the full disc is \( m \).
The mass of the cut portion is \( \frac{m}{16} \), as the radius of the cut portion is 5 cm and the original radius is 20 cm.
The center of mass of the full disc is at the origin, \( x_{\text{cm}} = 0 \).
The center of mass of the cut portion is at a distance of 15 cm from the origin, since the hole is cut from the edge of the disc.
To calculate the new center of mass: \[ X_{\text{com}} = \frac{m \times 0 - \frac{m}{16} \times 15}{m - \frac{m}{16}} = \frac{- \frac{m}{16} \times 15}{\frac{15m}{16}} = 1.0 \, \text{cm} \] Thus, the center of mass of the remaining disc is 1.0 cm from the origin.