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 ascertain the value of \(n\), the coordinates of the center of mass for the specified plate geometry must be computed. The plate comprises a rectangle with an embedded void, treatable as a composite of basic geometric forms. This composition allows for the application of the subtraction principle in determining the center of mass.

Step-by-Step Solution:

1. Shape Identification: The plate is constituted by a primary rectangle measuring 3 units by 2 units, from which a secondary rectangular section of 1 unit by 1 unit is removed.
2. Calculation of Total Plate Area:
   Area of the primary rectangle: \(3 \times 2 = 6 \, \text{unit}^2\)
   Area of the removed section: \(1 \times 1 = 1 \, \text{unit}^2\)
   Net area of the plate: \(6 - 1 = 5 \, \text{unit}^2\)
3. Determination of Center of Mass for Each Component:
   Primary Rectangle:
   Centroid coordinates: \((\frac{3}{2}, 1)\)
   Area: 6
   Removed Section:
   Centroid coordinates: \((1.5, 1.5)\)
   Area: 1
4. Application of Center of Mass Formula:
   For systems of combined bodies, the center of mass \((x_c, y_c)\) is determined by:
   \(x_c = \frac{\sum m_i x_i}{\sum m_i}\)
   \(y_c = \frac{\sum m_i y_i}{\sum m_i}\)
   Employing areas as mass equivalents, the calculation proceeds as follows:
   \(x_c = \frac{6 \times \frac{3}{2} - 1 \times 1.5}{6 - 1} = \frac{9 - 1.5}{5} = \frac{7.5}{5} = 1.5\)
   \(y_c = \frac{6 \times 1 - 1 \times 1.5}{6 - 1} = \frac{6 - 1.5}{5} = \frac{4.5}{5} = 0.9\)
5. Derivation of the Ratio:
   \(\frac{x_c}{y_c} = \frac{1.5}{0.9} = \frac{15}{9}\)
   Given the equivalence \(\frac{n}{9} = \frac{15}{9}\), it follows that \(n = 15\).

The derived value for \(n\) is 15, which is consistent with the provided range of 15,15.