Position of the Centre of Mass

Q 1 :
A uniform thin metal plate of mass 10 kg with dimensions is shown. The ratio of $x$ and $y$ coordinates of center of mass of plate is $\frac{n}{9}$ . The value of $n$ is _____. [2024]

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(15)

$\Rightarrow m_{1} x_{1} + m_{2} x_{2} = m_{3} x_{3}$

$10 x_{1} + 2 (1.5) = 12 (1.5) \Rightarrow x_{1} = 1.5 cm$

$\Rightarrow m_{1} y_{1} + m_{2} y_{2} = m_{3} y_{3}$

$10 y_{1} + 2 (1.5) = 12 \times 1 \Rightarrow y_{1} = 0.9 cm$

$\frac{x_{1}}{y_{1}} = \frac{1.5}{0.9} = \frac{15}{9} \Rightarrow n = 15$

Q 2 :
The identical spheres, each of mass 2M, are placed at the corners of a right-angled triangle with mutually perpendicular sides equal to 4 m each. Taking the point of intersection of these two sides as the origin, the magnitude of the position vector of the center of mass of the system is $\frac{4 \sqrt{2}}{x}$ , where the value of $x$ is ______. [2024]

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(3)

$Position vector {\vec{r}}_{C O M} = \frac{m_{1} {\vec{r}}_{1} + m_{2} {\vec{r}}_{2} + m_{3} {\vec{r}}_{3}}{m_{1} + m_{2} + m_{3}}$

${\vec{r}}_{C O M} = \frac{2 M \times 0 + 2 M \times 4 \hat{i} + 2 M \times 4 \hat{j}}{6 M}$

$\vec{r} = \frac{4}{3} \hat{i} + \frac{4}{3} \hat{j}$

$| \vec{r} |$ $= \sqrt{{(\frac{4}{3})}^{2} + {(\frac{4}{3})}^{2}} = \frac{4}{3} \sqrt{2} = \frac{4 \sqrt{2}}{x}$

$x = 3$

Q 3 :
Consider a circular disc of radius 20 cm with centre located at the origin. A circular hole of a radius 5 cm is cut from this disc in such a way that the edge of the hole touches the edge of the disc. The distance of centre of mass of residual or remaining disc from the origin will be: [2025]

2.0 cm
0.5 cm
1.5 cm
1.0 cm

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(4)

M $\propto$ A

$x_{c m} = \frac{m_{1} x_{1} + m_{2} x_{2}}{m_{1} + m_{2}}$

$= \frac{A (0) + (- \frac{A}{16}) (15)}{A - \frac{A}{16}}$

$= \frac{- 15}{16 - 1} = - 1 cm$

Q 4 :
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]

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

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

Q 5 :
A rod of length 5L is bent right angle keeping one side length as 2L. The position of the centre of mass of the system: (Consider L = 10 cm) [2025]

$2 \hat{i} + 3 \hat{j}$
$3 \hat{i} + 7 \hat{j}$
$5 \hat{i} + 8 \hat{j}$
$4 \hat{i} + 9 \hat{j}$

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(4)

$x_{com} = \frac{2 m (10) + 3 m (0)}{5 m} = 4 cm$

$y_{com} = \frac{2 m (0) + 3 m (15)}{5 m} = 9 cm$

${\vec{r}}_{com} = (4 \hat{i} + 9 \hat{j}) cm$

Topic Question Set

Q 1 :
A uniform thin metal plate of mass 10 kg with dimensions is shown. The ratio of $x$ and $y$ coordinates of center of mass of plate is $\frac{n}{9}$ . The value of $n$ is _____. [2024]

0%

0%

Q 3 :
Consider a circular disc of radius 20 cm with centre located at the origin. A circular hole of a radius 5 cm is cut from this disc in such a way that the edge of the hole touches the edge of the disc. The distance of centre of mass of residual or remaining disc from the origin will be: [2025]

Q 4 :
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]

Q 5 :
A rod of length 5L is bent right angle keeping one side length as 2L. The position of the centre of mass of the system: (Consider L = 10 cm) [2025]

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Topic Question Set

Q 1 : A uniform thin metal plate of mass 10 kg with dimensions is shown. The ratio of x and y coordinates of center of mass of plate is n9. The value of n is _____. [2024]

0%

0%

Q 3 : Consider a circular disc of radius 20 cm with centre located at the origin. A circular hole of a radius 5 cm is cut from this disc in such a way that the edge of the hole touches the edge of the disc. The distance of centre of mass of residual or remaining disc from the origin will be: [2025]

Q 4 : 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]

Q 5 : A rod of length 5L is bent right angle keeping one side length as 2L. The position of the centre of mass of the system: (Consider L = 10 cm) [2025]

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Q 1 :
A uniform thin metal plate of mass 10 kg with dimensions is shown. The ratio of $x$ and $y$ coordinates of center of mass of plate is $\frac{n}{9}$ . The value of $n$ is _____. [2024]

Q 3 :
Consider a circular disc of radius 20 cm with centre located at the origin. A circular hole of a radius 5 cm is cut from this disc in such a way that the edge of the hole touches the edge of the disc. The distance of centre of mass of residual or remaining disc from the origin will be: [2025]

Q 4 :
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]

Q 5 :
A rod of length 5L is bent right angle keeping one side length as 2L. The position of the centre of mass of the system: (Consider L = 10 cm) [2025]