Distance of the centroid of the leftover lamina from the original center A circular plate of diameter 6 cm (R = 3 cm) has a circular disc removed whose diameter equals the original radius (3 cm, so r = 1.5 cm). Find the distance of the centroid of the remaining lamina from the center of the original plate.

Introduction / Context:
Designers often remove material from plates to save weight while controlling balance. The centroid of the remaining area shifts toward the heavier side. Computing that shift ensures proper placement of supports, bearings, or counterweights.

Given Data / Assumptions:

• Original circle radius R = 3 cm (area A1 = 9π).
• Removed circle radius r = 1.5 cm (area A2 = 2.25π).
• The small circle is cut at the rim, internally tangent to the outer boundary → center spacing d = R − r = 1.5 cm.
• Uniform thickness and density (area centroid problem).

Concept / Approach:

Use the negative-area method along the line of centers. The centroid shift x from the original center equals the first moment of removed area divided by the net remaining area.

Step-by-Step Solution:

Verification / Alternative check:

By symmetry, the centroid must lie on the line of centers. Magnitude is less than 1.5 cm and proportional to area ratio A2/(A1 − A2), confirming 0.5 cm.

Why Other Options Are Wrong:

2.0 and 1.5 cm exceed d; 1.0 cm overestimates; 0.25 cm underestimates given the removed area fraction 2.25/6.75 = 1/3.

Common Pitfalls:

Confusing diameter and radius; forgetting to subtract the removed area in the denominator; placing the small circle incorrectly.

Final Answer:

0.5 cm