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

To determine the center of mass of the remaining portion of a circular disc after a smaller circular hole is removed, follow these steps:

1. Start with a circular disc of radius \( R = 20 \) cm, centered at the origin \((0, 0)\).
2. Cut a hole with a radius \( r = 5 \) cm such that its edge is tangent to the main disc's edge.
3. The distance between the center of the hole and the center of the original disc is \( R - r = 20 \, \text{cm} - 5 \, \text{cm} = 15 \, \text{cm} \).

Employ the formula for the center of mass of a composite body:

\(X_{\text{cm}} = \frac{A_1 X_1 + A_2 X_2}{A_1 + A_2}\)

• \(A_1\) = Area of the original disc = \(\pi R^2 = \pi (20)^2 = 400\pi \, \text{cm}^2\)
• \(A_2\) = Area of the hole = \(\pi r^2 = \pi (5)^2 = 25\pi \, \text{cm}^2\)
• The center of mass of the original disc (\(X_1\)) is at \((0, 0)\).
• The center of mass of the hole (\(X_2\)) is located at \((15, 0)\), given its tangency to the edge of the main disc.

Calculate the X-coordinate of the center of mass for the remaining area:

\(X_{\text{cm}} = \frac{400\pi \cdot 0 - 25\pi \cdot 15}{400\pi - 25\pi}\)

\(X_{\text{cm}} = \frac{-375\pi}{375\pi} = -1 \, \text{cm}\)

The center of mass is located 1 cm in the negative x-direction relative to the origin.

Therefore, the distance of the center of mass from the origin is 1.0 cm.

1.0 cm