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

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: $n = 15$.

Answer 2

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: $n = 15$.