Question

The center of mass of a thin rectangular plate (fig - x) with sides of length \( a \) and \( b \), whose mass per unit area (\( \sigma \)) varies as \( \sigma = \sigma_0 \frac{x}{ab} \) (where \( \sigma_0 \) is a constant), would be

• A. \( \left(\frac{2}{3} a, \frac{2}{3} b\right) \)
• B. \( \left(\frac{1}{3} a, \frac{1}{2} b\right) \)
• C. \( \left(\frac{1}{2} a, \frac{1}{2} b\right) \)
• D. \( \left(\frac{2}{3} a, \frac{1}{2} b\right) \)

Hint:

In problems involving variable density, set up the integral for each coordinate weighted by the density and normalized by the total mass.

Answer 1

The Correct Option is A

Using the variable density equation and integrating over the entire area, the \( x \)-coordinate of the center of mass \( \bar{x} \) is given by \( \int x \sigma \, dx \). After integration and applying the mass distribution, the resulting coordinates for \( \bar{x} \) and \( \bar{y} \) are \( \frac{2}{3} a \) and \( \frac{2}{3} b \) respectively.