Center of Mass as an Average Point

Q.

Define dimension, dimensional equation and dimension formula.

Q.

The height of mercury column in a barometer in a Calcutta laboratory was recorded to be 75 cm. Calculate this pressure in SI and CGS units using the following data : Specific gravity of mercuty = 13.6, Density of water = ${10}^{3}kg/m^3$, g = $9.8m/s^2$ at Calcutta. Pressure = hpg in usual symbols.

Q.

The linear mass density of a thin rod AB of length L varies from A to B as $\lambda \left(x\right)={\lambda }_0[1+(x/L\left)\right]$, Where $x$ is the distance from $A$. If $M$ is the mass of the rod then its moment of inertia about an axis passing through A and perpendicular to the rod is:

1. $\left(\frac{2}{5}\right){\mathrm{ML}}^2$

2. $\left(\frac{5}{12}\right){\mathrm{ML}}^2$

3. $\left(\frac{7}{18}\right)M{L}^2$

4. $\left(\frac{3}{7}\right)M{L}^2$

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. Centre of mass of a system of particles does not depend upon

1. forces acting on the particles
2. position of particles
3. relative distance between the particles
4. mass of the particles

Q. Three masses are placed on the x-axis: 300 g at origin, 500 g at $x=40$ cm and 400 g at $x=70\text{cm}$. The distance of the centre of mass from the origin

1. 50 cm
2. 30 cm
3. 45 cm
4. 40 cm

Q. A uniform metal disc of radius R is taken and out of it a disc of diameter R is cut off from the end. The centre of mass of the remaining part will be

1. $\frac{R}{4}$ from the centre
2. $\frac{R}{3}$ from the centre
3. $\frac{R}{5}$ from the centre
4. $\frac{R}{6}$ from the centre

Q.

Three equal masses m are placed at the three corners of an equilateral triangle of side a. Find the force exerted by this system on another particle of mass m placed at (a) the mid-point of a side, (b) at the centre of the triangle.

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.

In HCl molecule, the separation between the nuclei of hydrogen and chlorine atoms is $1.27\stackrel{\circ }{A}$. If the mass of a chlorine atom is 35.5 times that of a hydrogen atom, the centre of mass of the HCl molecule is at a distance of

1. $\frac{35.5\times 1.27}{36.5}\stackrel{\circ }{A}$ from the hydrogen atom.

2. $\frac{35.5\times 1.27}{36.5}\stackrel{\circ }{A}$ from the chlorine atom.

3. $\frac{1.27}{36.5}\stackrel{\circ }{A}$ from the hydrogen atom.

4. $\frac{1.27}{36.5}\stackrel{\circ }{A}$ from the chlorine atom.

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

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

Q. All the particles of a body are situated at a distance $R$ from the origin. The distance of the centre of mass of the body from the origin is

1. $=R$
2. $\le R$
3. $>R$
4. $\ge R$

Q.

Four particles of masses $m_1=2m,m_2=4m,m_3=m$ and $m_4$ respectively are placed at the four corners of a square. What should be the value of mass $m_4$ so that the centre of mass of the system of particles lies at the centre of square?

1. $m$
2. $2m$
3. $4m$
4. $6m$

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 silver atom in a solid oscillates in simple harmonic motion in some direction with a frequency of ${10}^{12}{s}^{-1}$. What is the force constant of the bonds connecting one atom with the other? (Mole wt. of silver = $108amu$ and $N_A=6.02x{10}^{23}gmo{l}^{-1}$)

1. $2.2N/m$

2. $5.5N/m$

3. $6.4N/m$

4. $7.1N/m$

Q. The position vector of three particles of masses $m_1=1\text{kg},m_2=2\text{kg}$ and $m_3=3\text{kg}$ are ${\overrightarrow{r}}_1=(\hat{i}+4\hat{j}+\hat{k})\text{m},{\overrightarrow{r}}_2=(\hat{i}+\hat{j}+\hat{k})\text{m}$ and ${\overrightarrow{r}}_3=(2\hat{i}-\hat{j}-2\hat{k})\text{m}$ respectively. The position vector of their centre of mass is

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

Q.

Does the centre of mass always lie within the body?

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. The distance between the carbon atom and the oxygen atom in a carbon monoxide molecule is 1.1 $\stackrel{o}{A}$. Given, mass of carbon atom is 12 a.m.u. and mass of oxygen atom is 16 a.m.u., calculate the position of the centre of mass of the carbon monoxide molecule

1. 0.63 from the carbon atom
2. 6.3 from the carbon atom
3. 1 from the oxygen atom
   
4. 0.12 from the oxygen atom

Q. A person of mass $60kg$ inside a lift of mass $940kg$ presses the button on control panel. The lift starts moving upwards with an acceleration of 1 $m/s^2$. If $g=10m/s^2$ , the tension in the supporting cable is

1. $10340N$
2. $660N$
3. $10000N$
4. $11000N$

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. 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. Find the value of $\lambda$.

1. $2.7$
2. $5.4$
3. $2.25$
4. $1.35$

Q. A rigid body can be hinged about any point on the x-axis. When it is hinged such that the hinge is at x, the moment of inertia is given by $I=2x^2–12x+27$. The x-coordinate of centre of mass is

Q.

Particles of masses m, 2m, 3m ........... nm grams are placed on the same line at distances, l, 2l, 3l, ...... nl cm from a fixed point. The distance of centre of mass of the particles from the fixed point in centimetres in

1. 

2. 

3. 

4. 

Q.

A light rod of length 1 m is pivoted at its centre and two masses of 5 kg and 2 kg are hung from the ends as shown in figure (10-E3). Find the initial angular acceleration of the rod assuming that it was horizontal in the beginning.

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

Q. Masses $8\text{kg}$, $2\text{kg}$ , $4\text{kg}$ & $2\text{kg}$ are placed at the corners $\text{A, B, C, D}$ respectively of a square $\text{ABCD}$ of diagonal $80\text{cm}$ (as shown in the figure below). The distance of the centre of mass from $A$ will be

1. $20\text{cm}$
2. $30\text{cm}$
3. $40\text{cm}$
4. $60\text{cm}$

Q.

According to einstein formula

M= m_{​​​​​​0^{/(1- v2/c2 ). As the body speeds up the mass will be greater.}}

^{_{so does that mean mass is not lost instead accumulated when a body speed up?}}

Q.

Can the centre of mass be outside an object?

Q. From a circular disc of radius R, a square is cut out with a radius as its diagonal. The centre of mass of remainder is at a distance (from the centre)

1. $\frac{R}{(4\pi -2)}$
2. $\frac{R}{2\pi }$
3. $\frac{R}{(\pi -2)}$
4. $\frac{R}{(2\pi -2)}$