Velocity of COM

Q. Two particles of mass $1\text{kg}$ and $3\text{kg}$ move towards each other under their mutual force of attraction. No other forces act on them. When the velocity of approach of the two particles is $2\text{m/s}$, their centre of mass has a velocity of $0.5\text{m/s}$. When the velocity of approach becomes $3\text{m/s}$, the velocity of the centre of mass of the system of particles is

1. $0.75\text{m/s}$
2. $0.5\text{m/s}$
3. $1.25\text{m/s}$
4. $1.5\text{m/s}$

Q. A system of particles is free from any external force. If $\overrightarrow{v}$ and $\overrightarrow{a}$ be the velocity and acceleration of the centre of mass, then it necessarily follows that

1. $\overrightarrow{v}=0;\overrightarrow{a}=0$
2. $\overrightarrow{v}\ne 0;\overrightarrow{a}=0$
3. $\overrightarrow{v}=0;\overrightarrow{a}\ne 0$
4. Both (a) and (b)

Q. Four particles of masses $1\text{kg},2\text{kg},3\text{kg}$ and $4\text{kg}$ situated at the corners of a square start moving at speeds of $8\text{m/s},6\text{m/s},4\text{m/s}$ and $2\text{m/s}$ respectively. Each particle maintains a direction towards the particle at the next corner symmetrically. Find the magnitude of velocity of the center of mass of the system at this instant.

1. $\frac{2}{5\sqrt{2}}\text{m/s}$
2. $\frac{5\sqrt{2}}{2}\text{m/s}$
3. $\frac{2\sqrt{2}}{5}\text{m/s}$
4. $\frac{\sqrt{2}}{5}\text{m/s}$

Q. Two bodies of masses of $1\text{kg}$ and $3\text{kg}$ initially at rest are moving with velocities $30\text{m/s}$ and $10\text{m/s}$ towards each other due to mutual force of attraction. What is the velocity of the centre of mass of system?

1. $8\text{m/s}$
2. $\text{Zero}$
3. $2\text{m/s}$
4. $4\text{m/s}$

Q. The curved part of the surface shown is semi-circular. All the surfaces are frictionless. Speed of the bigger block when the smaller block reaches the point $A$ of the surface is:

1. $\frac{Mv}{M+m}$
2. $\frac{mv}{M-m}$
3. $\frac{Mv}{M-m}$
4. $\frac{mv}{M+m}$

Q. In the 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. A light spring of force constant $K$ is held between two blocks of masses $m$ and $2m$. The two blocks and the spring system rests on a smooth horizontal floor. Now the blocks are moved towards each other compressing the spring by $x$ and then they are suddenly released. Then find the relative velocity between the blocks when the spring attains its natural length.

1. $\left(\sqrt{\frac{3K}{2m}}\right)x$
2. $\left(\sqrt{\frac{3K}{m}}\right)x$
3. $\left(\frac{K}{m}\right)x$
4. $\left(\sqrt{\frac{K}{2m}}\right)x$

Q. In a two block system shown in figure, match the following.



$\begin{array}{cc}\text{Table -1}& \text{Table -2}\\ \text{(A) Velocity of centre of}& \text{(P) Keep on changing all}\\ \text{mass}& \text{the time}\\ \text{(B) Momentum of centre of}& \text{(Q) First decreases then}\\ \text{mass}& \text{becomes zero}\\ \text{(C) Momentum of 1 kg block with respect to 2 kg block}& \text{(R) zero}\\ & \text{(S) constant}\end{array}$

1. $\text{A}\to \text{P},\text{B}\to \text{P, Q},\text{C}\to \text{Q}$
2. $\text{A}\to \text{P},\text{B}\to \text{R, S},\text{C}\to \text{Q}$
3. $\text{A}\to \text{R, S},\text{B}\to \text{R, S},\text{C}\to \text{R}$
4. $\text{A}\to \text{R, S},\text{B}\to \text{R, S},\text{C}\to \text{Q}$

Q.

Two particles A and B initially at rest, move towards each other by mutual force of attraction. At the instant when the speed of A is v and the speed of B is 2v, the speed of the centre of mass of the system is

1. 3v

2. v

3. 1.5 v

4. Zero

Q. Two particles of the same mass are projected from the ground with the same speed $10\text{m/s}$ at an angle ${37}^{\circ }$ and ${53}^{\circ }$ with the horizontal respectively. Find the maximum height attained by their $COM$ (in metres). Take $(g=9.8{\text{m/s}}^2)$ and answer upto two decimal places.

Q. Two boys $A$ and $B$ of mass $60\text{kg}$ and $40\text{kg}$ are standing on a platform. Both start walking towards each other with the same velocity $5\text{m/s}$. Find the velocity of the platform if mass of the platform is $100\text{kg}$.

1. $0.5\text{m/s}$
2. $2\text{m/s}$
3. $1\text{m/s}$
4. $5\text{m/s}$

Q. Two particles $A$ and $B$ of mass $1\text{kg}$ and $2\text{kg}$ are moving with velocity $u_A=10\text{m/s}$ and $u_B=20\text{m/s}$ respectively in opposite direction as shown in the figure. Find the magnitude of velocity of the centre of mass of the system.

1. $10\text{m/s}$
2. $15\text{m/s}$
3. $5\text{m/s}$
4. $20\text{m/s}$
   

Q. Two particles A and B initially at rest move towards each other under mutual force of attraction. At the instant when the velocity of A is $v$ and that of B is $2v$, the velocity of the center of mass of the system will be

1. $v$
2. $2v$
3. $3v$
4. $\text{Zero}$

Q. Two boys $A$ and $B$ of mass $60\text{kg}$ and $40\text{kg}$ are standing on a platform. Both start walking towards each other with the same velocity $5\text{m/s}$. Find the velocity of the platform if mass of the platform is $100\text{kg}$.

1. $0.5\text{m/s}$
2. $2\text{m/s}$
3. $1\text{m/s}$
4. $5\text{m/s}$

Q. A $200\text{kg}$ cannon at rest contains a $10\text{kg}$ cannon ball. When fired, the cannon ball leaves the cannon with a speed of $90\text{m/s}$. What is the recoil speed of the cannon?

1. $10\text{m/s}$
2. $5\text{m/s}$
3. $4.5\text{m/s}$
4. $8\text{m/s}$

Q. In a two block system shown in figure, match the following.



$\begin{array}{cc}\text{Table -1}& \text{Table -2}\\ \text{(A) Velocity of centre of}& \text{(P) Keep on changing all}\\ \text{mass}& \text{the time}\\ \text{(B) Momentum of centre of}& \text{(Q) First decreases then}\\ \text{mass}& \text{becomes zero}\\ \text{(C) Momentum of 1 kg block with respect to 2 kg block}& \text{(R) zero}\\ & \text{(S) constant}\end{array}$

1. $\text{A}\to \text{P},\text{B}\to \text{P, Q},\text{C}\to \text{Q}$
2. $\text{A}\to \text{P},\text{B}\to \text{R, S},\text{C}\to \text{Q}$
3. $\text{A}\to \text{R, S},\text{B}\to \text{R, S},\text{C}\to \text{R}$
4. $\text{A}\to \text{R, S},\text{B}\to \text{R, S},\text{C}\to \text{Q}$

Q. Two particles which are initially at rest, move towards each due to mutual attraction. If their speeds are $v$ and $2v$ at any instant, then the speed of centre of mass of the system will be

1. $2v$
2. $1.5v$
3. $0$
4. $v$

Q. Four bodies $A,B,C$ and $D$ of masses $[M_A=m,M_B=2m,M_C=m,M_D=2m]$ are initially at rest. They start moving under the influence of the internal force of attraction and at a particular instant, their velocities are $\overrightarrow{V_A}=2\hat{i},\overrightarrow{V_B}=-\hat{i}+2\hat{j}$ and $\overrightarrow{V_C}=-2\hat{i}+2\hat{j}$. Find the velocity of body $D\left(\overrightarrow{V_D}\right)$ at that instant.

1. $2\hat{i}-3\hat{j}$
2. $\hat{i}-3\hat{j}$
3. $\hat{i}+3\hat{j}$
4. $2\hat{i}+3\hat{j}$

Q. Two bodies of masses of $1\text{kg}$ and $3\text{kg}$ initially at rest are moving with velocities $30\text{m/s}$ and $10\text{m/s}$ towards each other due to mutual force of attraction. What is the velocity of the centre of mass of system?

1. $8\text{m/s}$
2. $\text{Zero}$
3. $2\text{m/s}$
4. $4\text{m/s}$

Q. Two spheres of mass $2\text{kg}$ and $6\text{kg}$ are moving with velocities $4\text{m/s}$ and $8\text{m/s}$ away from each other along the same line. The ball of mass $2\text{kg}$ is moving towards left. The velocity of the center of mass is

1. $4\text{m/s}$
2. $7\text{m/s}$
3. $5\text{m/s}$
4. $3\text{m/s}$

Q. A block of mass $m$ moving with a velocity $V_0$ collides with a stationary block of mass $M$ to which a spring of stiffness $k$ is attached, as shown in figure. Choose the correct alternative(s).

1. The velocity of centre of mass is $v_0$.
2. The initial kinetic energy of the system in the centre of mass frame is $\frac{1}{4}\left(\frac{mM}{m+M}\right)v_0^2$ .
3. The maximum compression in the spring is $v_0\sqrt{\frac{mM}{m+M}\frac{1}{k}}$ .
4. When the spring is in the state of maximum compression, the kinetic energy in the centre of mass frame is zero.

Q. Two bodies of mass $m$ are sliding down on the two sides of an equilateral triangular wedge of mass $10m$. Find the velocity of the wedge ($v$) when both the blocks reach the ground. (All surfaces are frictionless)

1. $v=0$
2. $v=g\hat{i}$
3. $v=-g\hat{i}$
4. $v=g\hat{j}$

Q. Two particles of the same mass move in the x - y plane with velocities $2\hat{i}{\text{ms}}^{-1}$ and $3\hat{j}{\text{ms}}^{-1}$. The first particle is given an acceleration $(2\hat{i}+2\hat{j}){\text{ms}}^{-2}$ and the second particle in given an acceleration $(2\hat{i}+4\hat{j}){\text{ms}}^{-2}$. The centre of mass of the system of two particles will move along a

1. Straight line
2. Circle
3. Ellipse
4. Parabola

Q. Two bodies of masses $10\text{kg}$ and $2\text{kg}$ are moving with velocities $2\hat{i}-7\hat{j}+3\hat{k}$ and $-10\hat{i}+35\hat{j}-3\hat{k}\text{m/s}$ respectively. The velocity of their centre of mass is

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

Q. A particle of mass $1\text{kg}$ has velocity ${\overrightarrow{v}}_1=\left(2t\right)\hat{i}$ and another particle of mass $2\text{kg}$ has velocity ${\overrightarrow{v}}_2=\left(t^2\right)\hat{j}$.

$\begin{array}{cc}\text{Columm I}& \text{Column II}\\ \text{(a) Net force on centre of mass at}2\text{s}& \text{(p)}\frac{20}{9}\text{unit}\\ \text{(b) Velocity of centre of mass at}2\text{s}& \text{(q)}\sqrt{68}\text{unit}\\ \text{(c) Displacement of centre of mass at}2\text{s}& \text{(r)}\frac{\sqrt{80}}{3}\text{unit}\\ & \text{(s) None}\end{array}$

1. a-q, b-r, c-p
2. a-r, b-q, c-p
3. a-r, b-p, c-r
4. a-r, b-p, c-r

Q. Four particles of masses $1\text{kg},2\text{kg},3\text{kg}$ and $4\text{kg}$ situated at the corners of a square start moving at speeds of $8\text{m/s},6\text{m/s},4\text{m/s}$ and $2\text{m/s}$ respectively. Each particle maintains a direction towards the particle at the next corner symmetrically. Find the magnitude of velocity of the center of mass of the system at this instant.

1. $\frac{2}{5\sqrt{2}}\text{m/s}$
2. $\frac{5\sqrt{2}}{2}\text{m/s}$
3. $\frac{2\sqrt{2}}{5}\text{m/s}$
4. $\frac{\sqrt{2}}{5}\text{m/s}$

Q. Two particles of different mass and uniform velocity are moving towards each other as shown in the figure. Find the displacement of the $COM$ after $5\text{s}$.

1. $50\text{m}$ towards left
2. $50\text{m}$ towards right
3. $10\text{m}$ towards left
4. $10\text{m}$ towards right

Q. A cannon of mass $m_1$ fires a shell of mass $m_2$ with speed $v_0$ relative to the barrel which is inclined at an angle $\alpha$ with horizontal . Find the recoil speed of cannon if it is placed on ice.

1. $\frac{m_2{v}_0\cos \alpha }{(m_1+m_2)}$
2. $\frac{m_1{v}_0\sin \alpha }{(m_1+m_2)}$
3. $\frac{m_2{v}_0\cos \alpha }{(m_1-m_2)}$
4. $\frac{m_1{v}_0\sin \alpha }{(m_1-m_2)}$

Q. Two masses $m_1$ and $m_2$ are moving with velocities $v_1$ and $v_2$ respectively. Find their total kinetic energy in the reference frame of centre of mass.

1. $\frac{m_1{m}_2}{m_1+m_2}(v_1-v_2{)}^2$
2. $\frac{1}{2}\frac{m_1{m}_2}{m_1+m_2}(v_1-v_2{)}^2$
3. $\frac{1}{2}\frac{m_1{m}_2}{m_1+m_2}(v_1+v_2{)}^2$
4. $\frac{1}{2}\left(m_1{m}_2\right)(v_1-v_2{)}^2$

Q. This section contains 1 Matrix Match type question, which has 2 Columns (Column I and Column II). Column I has four entries (A), (B), (C) and (D), Column II has four entries (P), (Q), (R) and (S). Match the entries in Column I with the entries in Column II. Each entry in Column I may match with one or more entries in Column II.
इस खण्ड में 1 मैट्रिक्स मिलान प्रकार का प्रश्न है, जिसमें 2 कॉलम (कॉलम I तथा कॉलम II) हैं। कॉलम I में चार प्रविष्टियाँ (A), (B), (C) तथा (D) हैं, कॉलम II में चार प्रविष्टियाँ (P), (Q), (R) तथा (S) हैं। कॉलम I में दी गयी प्रविष्टियों का मिलान कॉलम II में दी गयी प्रविष्टियों के साथ कीजिए। कॉलम I में दी गयी प्रत्येक प्रविष्टि का मिलान कॉलम II में दी गयी एक या अधिक प्रविष्टियों के साथ हो सकता है।

Match the Columns and choose the appropriate answer

कॉलमों का मिलान करें और सही उत्तर चुनें

A man of mass 60 kg is standing on a plank at one end, of mass 120 kg which in turn is placed on smooth horizontal surface. Man starts moving on plank with relative velocity 6 m/s. If length of plank is 18 m, then match the entries.

60 kg द्रव्यमान का एक व्यक्ति 120 kg द्रव्यमान के किसी तख्ते के एक सिरे पर खड़ा है तथा तख्ता चिकने क्षैतिज पृष्ठ पर रखा हुआ है। व्यक्ति, तख्ते पर 6 m/s के सापेक्षिक वेग से गति करना प्रारम्भ करता है। यदि तख्ते की लम्बाई 18 m है, तब प्रविष्टियों का मिलान कीजिए।

	Column (कॉलम)-I		Column (कॉलम)-II
(A)	Velocity of man (in m/s) व्यक्ति का वेग (m/s में)	(P)	2
(B)	Velocity of plank (in m/s) तख्ते का वेग (m/s में)	(Q)	3
(C)	Velocity of centre of mass (in m/s) of man and plank system व्यक्ति तथा तख्ते निकाय के द्रव्यमान केन्द्र का वेग (m/s में)	(R)	4
(D)	Time taken by man to reach other end (in s) व्यक्ति द्वारा दूसरे सिरे तक पहुंचने में लिया गया समय (s में)	(S)	Zero (शून्य)

1. A-R, B-P, C-S, D-Q
2. A-P, B-R, C-Q, D-S
3. A-Q, B-P, C-R, D-S
4. A-Q, B-R, C-S, D-P