Question

A uniform thin metal plate of mass \(10 \, \text{kg}\) with dimensions is shown. The ratio of \(x\) and \(y\) coordinates of the center of mass of the plate is \(\frac{n}{9}\). The value of \(n\) is ______.

Answer 1

Correct Answer: 15

The mass of the plate is calculated in three sections:
\(m1 = σ × 5 = 10 kg,\)
\(m2 = σ × 1 = 2 kg,\) 
\(m3 = σ × 6 = 12 kg.\)
Using the coordinates of each center of mass, we calculate the combined center of mass:
\[m_1x_1 + m_2x_2 = m_3x_3.\]
\[10 \cdot 1.5 + 2 \cdot (1.5) = 12 \cdot x_1 \implies x_1 = 1.5 \, \text{cm}.\]
Similarly:
\[m_1y_1 + m_2y_2 = m_3y_3.\]
\[10 \cdot 1 + 2 \cdot (1.5) = 12 \cdot y_1 \implies y_1 = 0.9 \, \text{cm}.\]
The ratio of $x_1$ to $y_1$ is:
\[\frac{x_1}{y_1} = \frac{1.5}{0.9} = \frac{15}{9}.\]
Thus:
\[n = 15.\]
Final Answer: $n = 15$.