Question

A thin rectangular plate of dimensions $a \times b$ lies in the xy-plane with one corner at the origin. The surface mass density at point $(x, y)$ is given by: $\sigma(x, y) = \sigma_0 \cdot \left(\frac{x^2 y}{a^2 b}\right)$ (where $\sigma_0$ is a constant)

Find the center of mass $(x_{CM}, y_{CM})$.

Answer 1

$\left(\frac{3a}{4}, \frac{2b}{3}\right)$

Answer 2

To find the center of mass $(x_{CM}, y_{CM})$ of a thin rectangular plate with a given surface mass density $\sigma(x,y)$, we use the following formulas:

$$x_{CM} = \frac{\iint x \, dm}{\iint dm} \quad \text{and} \quad y_{CM} = \frac{\iint y \, dm}{\iint dm}$$

Here, $dm = \sigma(x,y) \, dA = \sigma(x,y) \, dx \, dy$. The plate lies in the xy-plane with one corner at the origin and dimensions $a \times b$, so the integration limits are from $x=0$ to $x=a$ and $y=0$ to $y=b$. The surface mass density is given by $\sigma(x,y) = \sigma_0 \cdot \left(\frac{x^2 y}{a^2 b}\right)$.

1. Calculate the total mass (M):

$$M = \iint dm = \int_0^a \int_0^b \sigma_0 \frac{x^2 y}{a^2 b} \, dy \, dx$$ $$M = \frac{\sigma_0}{a^2 b} \int_0^a x^2 \left( \int_0^b y \, dy \right) \, dx$$

First, evaluate the inner integral:

$$\int_0^b y \, dy = \left[ \frac{y^2}{2} \right]_0^b = \frac{b^2}{2}$$

Substitute this back:

$$M = \frac{\sigma_0}{a^2 b} \int_0^a x^2 \left( \frac{b^2}{2} \right) \, dx = \frac{\sigma_0 b^2}{2a^2 b} \int_0^a x^2 \, dx = \frac{\sigma_0 b}{2a^2} \int_0^a x^2 \, dx$$

Now, evaluate the outer integral:

$$\int_0^a x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^a = \frac{a^3}{3}$$

Substitute this back to find M:

$$M = \frac{\sigma_0 b}{2a^2} \left( \frac{a^3}{3} \right) = \frac{\sigma_0 a b}{6}$$

2. Calculate the numerator for $x_{CM}$ ($\iint x \, dm$):

$$\iint x \, dm = \int_0^a \int_0^b x \cdot \sigma_0 \frac{x^2 y}{a^2 b} \, dy \, dx$$ $$\iint x \, dm = \frac{\sigma_0}{a^2 b} \int_0^a x^3 \left( \int_0^b y \, dy \right) \, dx$$

Using $\int_0^b y \, dy = \frac{b^2}{2}$:

$$\iint x \, dm = \frac{\sigma_0}{a^2 b} \int_0^a x^3 \left( \frac{b^2}{2} \right) \, dx = \frac{\sigma_0 b^2}{2a^2 b} \int_0^a x^3 \, dx = \frac{\sigma_0 b}{2a^2} \int_0^a x^3 \, dx$$

Now, evaluate the integral:

$$\int_0^a x^3 \, dx = \left[ \frac{x^4}{4} \right]_0^a = \frac{a^4}{4}$$

Substitute this back:

$$\iint x \, dm = \frac{\sigma_0 b}{2a^2} \left( \frac{a^4}{4} \right) = \frac{\sigma_0 a^2 b}{8}$$

3. Calculate $x_{CM}$:

$$x_{CM} = \frac{\iint x \, dm}{M} = \frac{\frac{\sigma_0 a^2 b}{8}}{\frac{\sigma_0 a b}{6}} = \frac{\sigma_0 a^2 b}{8} \cdot \frac{6}{\sigma_0 a b} = \frac{6a}{8} = \frac{3a}{4}$$

4. Calculate the numerator for $y_{CM}$ ($\iint y \, dm$):

$$\iint y \, dm = \int_0^a \int_0^b y \cdot \sigma_0 \frac{x^2 y}{a^2 b} \, dy \, dx$$ $$\iint y \, dm = \frac{\sigma_0}{a^2 b} \int_0^a x^2 \left( \int_0^b y^2 \, dy \right) \, dx$$

First, evaluate the inner integral:

$$\int_0^b y^2 \, dy = \left[ \frac{y^3}{3} \right]_0^b = \frac{b^3}{3}$$

Substitute this back:

$$\iint y \, dm = \frac{\sigma_0}{a^2 b} \int_0^a x^2 \left( \frac{b^3}{3} \right) \, dx = \frac{\sigma_0 b^3}{3a^2 b} \int_0^a x^2 \, dx = \frac{\sigma_0 b^2}{3a^2} \int_0^a x^2 \, dx$$

Using $\int_0^a x^2 \, dx = \frac{a^3}{3}$:

$$\iint y \, dm = \frac{\sigma_0 b^2}{3a^2} \left( \frac{a^3}{3} \right) = \frac{\sigma_0 a b^2}{9}$$

5. Calculate $y_{CM}$:

$$y_{CM} = \frac{\iint y \, dm}{M} = \frac{\frac{\sigma_0 a b^2}{9}}{\frac{\sigma_0 a b}{6}} = \frac{\sigma_0 a b^2}{9} \cdot \frac{6}{\sigma_0 a b} = \frac{6b}{9} = \frac{2b}{3}$$

The center of mass is $\left( \frac{3a}{4}, \frac{2b}{3} \right)$.

The final answer is $\boxed{\left(\frac{3a}{4}, \frac{2b}{3}\right)}$.

Explanation of the solution:

The center of mass $(x_{CM}, y_{CM})$ for a continuous two-dimensional object is found by integrating $x \, dm$ and $y \, dm$ over the entire mass and dividing by the total mass $M$. The differential mass element $dm$ is given by $\sigma(x,y) \, dx \, dy$. We performed double integrals for $M$, $x_{CM}$ numerator, and $y_{CM}$ numerator, substituting the given density function $\sigma(x,y)$ and the integration limits corresponding to the plate's dimensions.