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

To find the value of \(n\), we need to calculate the coordinates of the center of mass for the given plate shape. The shape is composed of a rectangle with a cut-out, which can be analyzed as a combination of simple geometrical figures, allowing us to use subtraction to find the center of mass.

Step-by-Step Solution:

1. Identify Shapes: The plate consists of a large rectangle (3 × 2) with a smaller rectangular cut-out (1 × 1).
2. Calculate Total Area of the Plate:
   Large Rectangle Area: \(3 \times 2 = 6 \, \text{unit}^2\)
   Cut-out Area: \(1 \times 1 = 1 \, \text{unit}^2\)
   Total Area: \(6 - 1 = 5 \, \text{unit}^2\)
3. Calculate Center of Mass for Each Shape:
   Large Rectangle:
   Center: \((\frac{3}{2}, 1)\)
   Area: 6
   Cut-out:
   Center: \((1.5, 1.5)\)
   Area: 1
4. Apply the Center of Mass Formula:
   For combined objects, center of mass \((x_c, y_c)\) is:
   \(x_c = \frac{\sum m_i x_i}{\sum m_i}\)
   \(y_c = \frac{\sum m_i y_i}{\sum m_i}\)
   Using the areas as weights, calculate:
   \(x_c = \frac{6 \times \frac{3}{2} - 1 \times 1.5}{6 - 1} = \frac{9 - 1.5}{5} = \frac{7.5}{5} = 1.5\)
   \(y_c = \frac{6 \times 1 - 1 \times 1.5}{6 - 1} = \frac{6 - 1.5}{5} = \frac{4.5}{5} = 0.9\)
5. Determine the Ratio:
   \(\frac{x_c}{y_c} = \frac{1.5}{0.9} = \frac{15}{9}\)
   Here, \(\frac{n}{9} = \frac{15}{9}\), so \(n = 15\).

The calculated value of \(n\) is 15, which falls within the given range of 15,15.

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: $n = 15$.