Question

The coordinates of the centre of mass of a uniform L-shaped plate of mass 3 kg shown in the figure is:

 

• A. \( \left( \frac{5}{6} \, \text{m}, \frac{5}{6} \, \text{m} \right) \)
• B. \( \left( \frac{3}{2} \, \text{m}, \frac{3}{2} \, \text{m} \right) \)
• C. \( \left( \frac{1}{2} \, \text{m}, \frac{1}{2} \, \text{m} \right) \)
• D. \( \left( \frac{6}{5} \, \text{m}, \frac{6}{5} \, \text{m} \right) \)

Hint:

To find the center of mass of a composite body, break it into simpler shapes, compute their centroids and masses (proportional to area for uniform density), and apply the weighted average formula.

Answer 1

The Correct Option is A

Step 1: Divide the L-shape into two rectangles.
Let’s denote:
Block A: vertical rectangle of dimension \(1 \times 2\)
Block B: horizontal rectangle of dimension \(1 \times 1\)
Assume uniform density. Total mass = 3 kg. So, each block has:
Area of A = \(1 \times 2 = 2\) units,
Area of B = \(1 \times 1 = 1\) unit,
Total area = 3 units.
So, mass of A = 2 kg, mass of B = 1 kg.
Step 2: Coordinates of individual centroids
Centroid of A is at \( (0.5, 1) \)
Centroid of B is at \( (1.5, 0.5) \)
Step 3: Use the center of mass formula:
\[ x_{\text{cm}} = \frac{m_1x_1 + m_2x_2}{m_1 + m_2} = \frac{2(0.5) + 1(1.5)}{3} = \frac{1 + 1.5}{3} = \frac{2.5}{3} = \frac{5}{6} \] \[ y_{\text{cm}} = \frac{m_1y_1 + m_2y_2}{m_1 + m_2} = \frac{2(1) + 1(0.5)}{3} = \frac{2 + 0.5}{3} = \frac{2.5}{3} = \frac{5}{6} \] So, the center of mass = \( \left( \frac{5}{6} \, \text{m}, \frac{5}{6} \, \text{m} \right) \)