The centre of mass of a thin rectangular plate (fig) with sides of length a and b, whose mass per unit area () varies as (where is a constant), would be _____

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Q.
The centre of mass of a thin rectangular plate (fig) with sides of length a and b, whose mass per unit area ( $σ$ ) varies as $σ = \frac{σ_{0} x}{a b}$ (where $σ_{0}$ is a constant), would be ______. [2025]

1 $(\frac{2}{3} a, \frac{b}{2})$

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

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

4 $(\frac{1}{3} a, \frac{b}{2})$

Ans.
(1)

$σ$ is constant in y-direction, so, $y_{cm} = b / 2$

$x_{cm} = \frac{\int_{0}^{a} x d m}{\int_{0}^{a} d m} = \frac{\int_{0}^{a} x σ_{x} d A}{\int_{0}^{a} σ_{x} d A} = \frac{\int_{0}^{a} x \frac{σ_{0} x}{a b} b d x}{\int_{0}^{a} \frac{σ_{0} x}{a b} b d x}$

$x_{cm} = \frac{\int_{0}^{a} x^{2} d x}{\int_{0}^{a} x d x} = \frac{{(\frac{x^{3}}{3})}_{0}^{a}}{{(\frac{x^{2}}{2})}_{0}^{a}} = \frac{a^{3} / 3}{a^{2} / 2} = \frac{2 a}{3}$

i.e., ${\vec{r}}_{com} \equiv (\frac{2 a}{3}, \frac{b}{2})$

Solution

Q.
The centre of mass of a thin rectangular plate (fig) with sides of length a and b, whose mass per unit area ( $σ$ ) varies as $σ = \frac{σ_{0} x}{a b}$ (where $σ_{0}$ is a constant), would be ______. [2025]

1 $(\frac{2}{3} a, \frac{b}{2})$

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

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

4 $(\frac{1}{3} a, \frac{b}{2})$

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Solution

Q. The centre of mass of a thin rectangular plate (fig) with sides of length a and b, whose mass per unit area (σ) varies as σ=σ0xab (where σ0 is a constant), would be ______. [2025]

1 (23a,b2)

2 (23a,23b)

3 (a2,b2)

4 (13a,b2)

Ans. (1) σ is constant in y-direction, so, ycm=b/2 xcm=∫0axdm∫0adm=∫0axσxdA∫0aσxdA=∫0axσ0xabbdx∫0aσ0xabbdx xcm=∫0ax2dx∫0axdx=(x33)0a(x22)0a=a3/3a2/2=2a3 i.e., r→com≡(2a3,b2)

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Q.
The centre of mass of a thin rectangular plate (fig) with sides of length a and b, whose mass per unit area ( $σ$ ) varies as $σ = \frac{σ_{0} x}{a b}$ (where $σ_{0}$ is a constant), would be ______. [2025]

1 $(\frac{2}{3} a, \frac{b}{2})$

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

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

4 $(\frac{1}{3} a, \frac{b}{2})$