Center of Mass as an Average Point

Q. Three particles of masses $1\text{kg},2\text{kg}$ and $3\text{kg}$ are situated at the corners of an equilateral triangle of side $b$. The $(x,y)$ coordinates respectively for the centre of mass of the system of particles will be:

1. $[\frac{7b}{12},\frac{\sqrt{3}b}{12}]$
2. $[\frac{3\sqrt{3}b}{12},\frac{7b}{12}]$
3. $[\frac{7b}{12},\frac{3\sqrt{3}b}{12}]$
4. $[\frac{7b}{12},\frac{3\sqrt{3}b}{4}]$

Q. Which of the following statements are correct?
$1.$ Centre of mass of a body always coincides wiith the centre of gravity of the body.
$2.$ Centre of gravity of a body is the point where total gravitational torque on the body is zero.
$3.$ A couple on a body produces both translational and rotational motion in a body.
$4.$ Mechanical advantage greater than one means that a small effort can be used to lift a large load.

1. $\left(1\right)$ and $\left(2\right)$
   
2. $\left(2\right)$ and $\left(3\right)$
   
3. $\left(3\right)$ and $\left(4\right)$
   
4. $\left(2\right)$ and $\left(4\right)$
   

Q.

Four particles of masses m, m, 2m and 2m are placed at the four corners of a square of side a as shown in the figure. The (x, y) coordinates of the centre of mass are

1. $(\frac{a}{2},2a)$

2. $(\frac{a}{2},a)$

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

4. $(a,\frac{a}{3})$

Q. The position vector of four particles of masses $m_1=2\text{kg},m_2=4\text{kg},m_3=6\text{kg},m_4=8\text{kg}$ are $\overrightarrow{r_1}=2\hat{i}+3\hat{j}+0\hat{k},\overrightarrow{r_2}=0\hat{i}+2\hat{j}+3\hat{k},\overrightarrow{r_3}=2\hat{i}+0\hat{j}+3\hat{k};\overrightarrow{r_4}=2\hat{i}+3\hat{j}+3\hat{k}$. The position vector of their centre of mass is given by

1. $\frac{16\hat{i}+19\hat{j}+27\hat{k}}{11}$
2. $1.6\hat{i}+1.9\hat{j}+2.7\hat{k}$
3. $8\hat{i}+9\hat{j}+13\hat{k}$
4. $\frac{32\hat{i}+38\hat{j}+54\hat{k}}{11}$

Q. What is the position of Center of Mass for irregular shaped objects?

1. At any point on the surface.
2. At the Geometric Center
3. Closer to the more Massive End.
4. Closer to the least Massive End.

Q. Which of the following points is the likely position of the centre of mass of the system shown in the figure ?

1. A
2. B
3. C
4. D

Q. Three identical spheres each of mass $1\text{kg}$ are kept as shown in the figure. They are kept in such a way that they are touching each other with their respective centres on a straight line . If their centres are marked $P,Q,R$ respectively, then the distance of the centre of mass of the system from $P$ is:

1. $\frac{PQ+QR+PR}{3}$
2. $\frac{PQ+PR}{3}$
3. $\frac{PQ+PR}{2}$
4. $\frac{PQ+QR}{2}$

Q. Centre of mass of a system of three particles of masses $1\text{kg}$, $2\text{kg}$ and $3\text{kg}$ at the point $(1\text{m},2\text{m},3\text{m})$ and centre of mass of another group of two particles of masses $2\text{kg}$ and $3\text{kg}$ is at point $(-1\text{m},-2\text{m})$, where a $10\text{kg}$ particle should be placed, so that the centre of mass of the system of all those six particles shifts to centre of mass of the first system?

1. $(3\text{m},6\text{m},6\text{m})$
2. $(3\text{m},6\text{m},4\text{m})$
3. $(2\text{m},4\text{m},4.5\text{m})$
4. $(3\text{m},6\text{m},4.5\text{m})$

Q. Three particles of the same mass lie on the coordinates (1, 1), (2, 2) and (3, 3) respectively. The coordinates of their center of mass are (2, 2).

1. True
2. False

Q. The position vector of three particles of mass $m_1=3\text{kg},m_2=4\text{kg}$ and $m_3=1\text{kg}$ are $\overrightarrow{r_1}=(2\hat{i}+\hat{j}+3\hat{k})\text{m},\overrightarrow{r_2}=(\hat{i}-3\hat{j}+2\hat{k})\text{m}$ and $\overrightarrow{r_3}=(3\hat{i}-2\hat{j}-\hat{k})\text{m}$ respectively. Find the position vector of centre of mass of the system of particles.

1. $\frac{1}{8}(13\hat{i}-12\hat{j}+15\hat{k})\text{m}$
2. $\frac{1}{8}(11\hat{i}-13\hat{j}+16\hat{k})\text{m}$
3. $\frac{1}{8}(16\hat{i}+11\hat{j}-13\hat{k})\text{m}$
4. $\frac{1}{8}(13\hat{i}-11\hat{j}+16\hat{k})\text{m}$

Q.

Four particles, each of mass m, are placed at the corners of a square of side a as shown in the figure. The position vector of the centre of mass is

1. $a(\hat{i}+\hat{j})$

2. $\frac{a}{2}(\hat{i}+\hat{j})$

3. $a(\hat{i}-\hat{j})$

4. $\frac{a}{2}(\hat{i}-\hat{j})$

Q. Three particles $A,B$ and $C$ of masses $2\text{kg}$, $3\text{kg}$ and $5\text{kg}$ are placed at the vertices of a scalene triangle as shown in the figure. Find the location of $COM$ of the three particle system

1. $(\frac{4}{5},\frac{9}{10})\text{m}$
2. $(\frac{9}{10},\frac{4}{5})\text{m}$
3. $(\frac{4}{5},\frac{4}{5})\text{m}$
4. $(\frac{9}{10},\frac{9}{10})\text{m}$

Q. Three bodies of masses $1\text{kg}$, $3\text{kg}$ and $4\text{kg}$ have position vectors $(\hat{i}+2\hat{j}+\hat{k})$, $(-3\hat{i}-2\hat{j}+\hat{k})$ and $\left(4\hat{i}\right)$ respectively . The centre of mass of this system has position vector

1. $\frac{(2\hat{i}+\hat{j}-\hat{k})}{2}$
2. $\frac{(2\hat{i}+\hat{j}+\hat{k})}{2}$
3. $\frac{(2\hat{i}-\hat{j}-\hat{k})}{2}$
4. $\frac{(2\hat{i}-\hat{j}+\hat{k})}{2}$

Q. In the $\text{HCl}$ molecule, the separation between the nuclei of the two atoms is about $1.5\stackrel{\circ }{A}(1\stackrel{\circ }{A}={10}^{-10}\text{m})$. The approximate location of the centre of mass from the hydrogen atom assuming the chlorine atom to be about 35.5 times as massive as hydrogen is

1. $1.45{\text{A}}^{\circ }$
2. $0.05{\text{A}}^{\circ }$
3. $0.72{\text{A}}^{\circ }$
4. $0.96{\text{A}}^{\circ }$

Q.

A uniform rod of length $l=100cm$ is bent at its mid-point to make ${90}^{\circ }$ angle. The distance of the centre of mass from the centre of the rod is

1. 36.1 cm

2. 25.2 cm

3. 17.7 cm

4. Zero

Q. 3 particles of masses $2\text{kg}$ each, are placed such that $1^{st}$ one lies on negative $x-$ axis, $2^{nd}$ one lies on negative $y-$ axis and the third one lies on positive $z-$ axis at distances of $2\text{m},3\text{m}$ and $1\text{m}$ respectively from the origin. Then the square of the distance of centre of mass of the system from the origin is

1. $1.55{\text{m}}^2$
2. $\sqrt{1.55}{\text{m}}^2$
3. $1.55\text{m}$
4. $1.25\text{m}$

Q. Two particles of masses $m_1$ and $m_2$ are located at $x=0\text{m}$ and $x=6\text{m}$. If the position of their centre of mass is at $x=4\text{m}$, then the ratio of their masses $\frac{m_2}{m_1}$ is

1. $2$
2. $1$
3. $4$
4. $3$

Q. Four particles of masses $m_1,m_2,m_3$ and $m_4$ are placed at the vertices A, B, C and D of a square as shown respectively. The COM of the system will lie on diagonal AC if:

1. $m_1=m_4$
2. $m_2=m_4$
3. $m_1=m_2$
4. $m_3=m_4$

Q. Three particles of masses $1\text{kg},2\text{kg}$ and $3\text{kg}$ are situated at the corners of an equilateral triangle of side $b$. The $(x,y)$ coordinates respectively for the centre of mass of the system of particles will be:

1. $[\frac{7b}{12},\frac{\sqrt{3}b}{12}]$
2. $[\frac{3\sqrt{3}b}{12},\frac{7b}{12}]$
3. $[\frac{7b}{12},\frac{3\sqrt{3}b}{12}]$
4. $[\frac{7b}{12},\frac{3\sqrt{3}b}{4}]$

Q. Three boys are standing on a horizontal paltform of mass $170\text{kg}$ and length $6\text{m}$ is as shown in figure. They exchange their position as shown in the figure. Distance moved by the platform is : (Neglect friction)

1. $0.35\text{m}$
2. $0.55\text{m}$
3. $0.45\text{m}$
4. $0.25\text{m}$

Q. An isolated particle of mass $m$ is moving in horizontal plane $x-y$, along the x-axis, at a certain height above ground. It suddenly explodes into two fragments of masses $\frac{m}{4}$ and $\frac{3m}{4}$ . An instant later, the smaller fragment is at $y=+15\text{cm}$. The larger fragment at this instant is at

1. $y=-5\text{cm}$
2. $y=+5\text{cm}$
3. $y=-50\text{cm}$
4. $y=-20\text{cm}$

Q. The average weight of 8 person's increases by 2.5 kg when a new person comes in place of one of them weighing 65 kg. What might be the weight of the new person?

1. 76 kg
2. 80.5 kg
3. 85 kg
4. 92 kg

Q. If COM of a system of particles lies at the origin of the coordinate system then

1. x - coordinate of all the particles may be positive
2. x - coordinate of all the particles may be negative
3. x - coordinate of all the particles may be zero
4. y - coordinate of all the particles may be positive

Q. Find the x coordinate of the centre of mass of the arrangement of three uniform rods kept in horizontal plane, as shown in figure. (Consider mass and length of each rod is m and l respectively)

चित्र में दर्शाए अनुसार क्षैतिज तल में रखी एक जैसी तीन छड़ों के समंजन के द्रव्यमान केन्द्र का x निर्देशांक ज्ञात कीजिए। (माना प्रत्येक छड़ का द्रव्यमान तथा लम्बाई क्रमशः m तथा l हैं)

1. $\frac{2l}{3}$
2. $\frac{6l}{5}$
3. $\frac{9l}{7}$
4. $\frac{13l}{12}$

Q. 3 particles of masses $2\text{kg}$ each, are placed such that $1^{st}$ one lies on negative $x-$ axis, $2^{nd}$ one lies on negative $y-$ axis and the third one lies on positive $z-$ axis at distances of $2\text{m},3\text{m}$ and $1\text{m}$ respectively from the origin. Then the square of the distance of centre of mass of the system from the origin is

1. $1.55{\text{m}}^2$
2. $\sqrt{1.55}{\text{m}}^2$
3. $1.55\text{m}$
4. $1.25\text{m}$

Q. The centre of mass of a system of three particles of masses $2\text{g},3\text{g}$ and $5\text{g}$ is taken as the origin of a co-ordinate system. The position vector of a fourth particle of mass $8\text{g}$, such that the centre of mass of the four particle system lies at the point $(3,4,7)\text{m}$, is $\lambda (3\hat{i}+4\hat{j}+7\hat{k})\text{m}$ where $\lambda$ is a constant. Then, the value of $\lambda =$
(Answer upto two decimal places)

Q. Two bodies of masses $2\text{kg}$ & $5\text{kg}$ have position vectors $\hat{i}-2\hat{j}+\hat{k}$ and $3\hat{i}+2\hat{j}-\hat{k}$ respectively. The centre of mass of the system has a position vector

1. $2\hat{i}+4\hat{j}-3\hat{k}$
2. $\left(\frac{4\hat{i}+5\hat{j}-3\hat{k}}{7}\right)$
3. $17\hat{i}+6\hat{j}-3\hat{k}$
4. $\left(\frac{17\hat{i}+6\hat{j}-3\hat{k}}{7}\right)$

Q. Three bodies of masses $1\text{kg}$, $3\text{kg}$ and $4\text{kg}$ have position vectors $(\hat{i}+2\hat{j}+\hat{k})$, $(-3\hat{i}-2\hat{j}+\hat{k})$ and $\left(4\hat{i}\right)$ respectively . The centre of mass of this system has position vector

1. $\frac{(2\hat{i}+\hat{j}-\hat{k})}{2}$
2. $\frac{(2\hat{i}+\hat{j}+\hat{k})}{2}$
3. $\frac{(2\hat{i}-\hat{j}-\hat{k})}{2}$
4. $\frac{(2\hat{i}-\hat{j}+\hat{k})}{2}$

Q. Three particles of masses $1\text{kg}$ , $2\text{kg}$ and $3\text{kg}$ are placed at the vertices $A,B,C$ respectively of an equilateral triangle $ABC$ of edge $1\text{m}$ as shown in the figure. Find the position of the centre of mass of the system. (If $A$ is assumed to be the origin).

1. $(\frac{2}{3},\frac{\sqrt{3}}{6})$
2. $(\sqrt{\frac{3}{16}},\frac{2}{3})$
3. $(\frac{\sqrt{3}}{6},\frac{2}{3})$
4. $(\frac{2}{3},\sqrt{\frac{3}{16}})$

Q. A circular ring of mass $6\text{kg}$ and radius a is placed such that its centre lies at the origin. Two particles of masses $2\text{kg}$ each are placed at the intersecting points of the circle with +ve x-axis and +ve y -axis. Then, the angle made by the position vector of centre of mass of entire system with x - axis is

1. ${45}^{\circ }$
2. ${60}^{\circ }$
3. $ta{n}^{-1}\left(\frac{5}{4}\right)$
4. ${30}^{\circ }$