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# Math Quiz Trial

1. Choose the correct statement.

(a) A rigid body has perfectly definite shape

(b) Distance between any pair of particles of a rigid body shall not change

(c) No real body can be truly rigid

(d) All the above.

(NCERT Based)

2. A rigid body may have

(a) Pure Translational motion only

(b) Pure Rotational motion only

(c) Combination of Translational and Rotational motion

(d) All the above

(NCERT Based)

3. The motion of a rigid body, which is not fixed/pivoted is

(a) pure rotation

(b) pure translation

(c) combination of translation and rotation both

(d) both $(b)$ and $(c)$.

(NCERT Based)

4. When a cylinder rolls down an inclined plane, its motion is

(a) translational only

(b) rotational only

(c) combination of translation and rotation

(d) none of the above.

(NCERT Based)

5. Choose the incortect statement of the following :

(a) A single particle is treated as a point mass

(b) A single particle has no size and no shape

(c) A rigid body consists of a system of particles

(d) None of the above

(NCERT Based)

6. Choose the statement, which is incorrect.

(a) Any number of particles interacting with one another are said to form a system

(b) A system is a collection of particles which are non-interacting

(c) Any object of finite size can be regarded as a system

(d) None of the above

(NCERT Based

7. Out of the following, choose the correct statement.

(a) The forces exerted by various particles of the system on one another are called internal forces

(b) Though internal forces are mutual, they do not cancel one anothex

(c) Internal forces can produce motion in a body

(d) Internal forces can stop a moving body

(NCERT Based)

6.2 CENTRE OR MASS

1. In the $\mathrm{HCl}$ molecule, the separation between the nuclei of the two atoms is about $1.27 \AA\left(1 \AA^{1}=10^{-10} \mathrm{~m}\right)$. The approximate location of the centre of mass of the molecule from hydrogen atom assuming the chlorine atom to be about $35.5$ times massive as hydrogen is

(a) $1 \AA$

(b) $2.5 \mathrm{~A}$

(c) $1.24 \AA$

(d) $1.5 \AA$ (Kerala PET 2002)

2. Three identical spheres, each of mass $M$ are placed at the corners of a right angled triangle with mutually perpendicular sides equal to $2 m$ each. Taking their point of intersection as the origin, the position vector of centre of mass is

(a) $\frac{1}{3}(\hat{i}-\hat{j})$

(b) $\frac{2}{3}(\hat{i}-\hat{j})$

(c) $\frac{2}{3}(\hat{i}+\hat{j})$

(d) $\frac{1}{3}(\hat{i}+\hat{j})$

4. Two bodies of mass $1 \mathrm{~kg}$ and $3 \mathrm{~kg}$ 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 this system has a position vector

(a) $-\hat{i}+\hat{j}+\hat{k}$

(b) $-2 \hat{i}+2 \hat{k}$

(c) $-2 \hat{i}-\hat{j}+\hat{k}$

(d) $2 \hat{i}-\hat{j}-2 \hat{k}$

5. Centre of mass of 3 bodies $10 \mathrm{~kg}, 20 \mathrm{~kg}$ and $30 \mathrm{~kg}$ is at $(0,0,0)$. Where should a body of mass $40 \mathrm{~kg}$ be placed so that the combination centre of mass will be it $(3,3,3)$

(a) $(0,0,0)$

(b) $(7.5,7.5,7.5)$

(c) $(1,2,3)$

(d) $(4,4,4)$ (J \& $\mathrm{K}$ CET 2006$)$

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

(a) $40 \mathrm{~cm}$

(b) $45 \mathrm{~cm}$

(c) $50 \mathrm{~cm}$

(d) $30 \mathrm{~cm}$

9. Four point masses $P, Q, R$ and $S$ with respective masses $1 \mathrm{~kg}$, $1 \mathrm{~kg}, 2 \mathrm{~kg}$ and $2 \mathrm{~kg}$ form the corners of a square of side $a$. The centre of masses of the system will be farthest from

(a) $P$ only

(b) $R$ and $S$

(c) $R$ only

(d) $P$ and $Q$ (Kerala PET 2007)

10. The separation between $C$ and $O$ atoms in $C O$ is $1.2$ A. The distance of carbon atom from centre of mass is : assume mass of $C=14$ and mass of $O=16$

(a) $0.3 \AA$

(b) $0.64 \AA$

(c) $0.5 \AA$

(Odisha JEE 2002)

11. Two bodies of masses $1 \mathrm{~kg}$ and $3 \mathrm{~kg}$ 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 this system has a position vector

(a) $-2 \hat{i}+2 \hat{k}$

(b) $-2 \hat{i}-\hat{j}+\hat{k}$

(c) $2 \hat{i}-\hat{j}-\hat{k}$

(d) $-\hat{i}+\hat{j}+\hat{k}$

17. Where will be the centre of mass on combining two masses $m$ and $M(M>m)$

(a) towards $m$

(b) towards $M$

(c) between $m$ and $M$

(d) anywhere

(RPET 2003)

18. A rod of mass $m$ and length $l$ is made to stand at an angle of $60^{\circ}$ with the vertical. Potential energy of the rod in this position is

(a) $\mathrm{mgl}$

(b) $\frac{m g l}{2}$

(c) $\frac{m g l}{3}$

(d) $\frac{m g l}{4}$

(e) $\frac{m g l}{\sqrt{2}}$

(Kerala PET 2009 )

19!. The centre of mass of a body

(a) lies always outside the body

(b) may lie within, outside or on the surface of the body

(c) lies always inside the body

(d) lies always on the surface of the body

(MH CET Med. 2001; MP PET 2012)

20. The centre of mass of a system of two particles divides the distance between them

(a) in inverse ratio of square of masses of particles

(b) in direct ratio of square of masses of particles

(c) in inverse ratio of masses of particles

(d) in direct ratio of masses of particles

(MH CET 2004)

1. Two point objects of masses $1.5 \mathrm{~g}$ and $2.5 \mathrm{~g}$ respectively are at a distance $16 \mathrm{~cm}$ apart, the centre of gravity is at a distance $x$ from the object of mass $1.5 \mathrm{~g}$, where $x$ is

(a) $10 \mathrm{~cm}$

(c) $13 \mathrm{~cm}$

(b) $6 \mathrm{~cm}$

(d) $3 \mathrm{~cm}$

23. A rod of length $3 \mathrm{~m}$ has its mass acting per unit length directly proportional to distance $x$ from one of its ends. The centre of mass of the rod from that end will be at

(a) $1.5 \mathrm{~m}$

(b) $2 \mathrm{~m}$

(c) $2.5 \mathrm{~m}$

(d) $3.0 \mathrm{~m}$

(A IPMT 2002)

6.3 MOTION OF CENTRE OF MASS

1. Two spherical bodies of mass $M$ and $5 M$ and radii $R$ and $2 R$ respectively are released in free space with initial separation between their centres equal to $12 R$. If they attract each other due to gravitational force only, then the distance covered by the smaller body just before collision is

(a) $1.5 \mathrm{R}$

(b) $2.5 \mathrm{R}$

(c) $4.5 R$

(d) $7.5 \mathrm{R}$

(AIEEE 2003)

2. 2 bodies of different masses of $2 \mathrm{~kg}$ and $4 \mathrm{~kg}$ are moving with velocities $20 \mathrm{~m} / \mathrm{s}$ and $10 \mathrm{~m} / \mathrm{s}$ towards each other due to mutual gravitational attraction. What is the velocity of their centre of mass

(a) $5 \mathrm{~m} / \mathrm{s}$

(b) $6 \mathrm{~m} / \mathrm{s}$

(c) $8 \mathrm{~m} / \mathrm{s}$

(d) zero

6. Identify the correct statement for the rotational motion of a rigid body

$(a)$ individual particles of the body do not undergo accelerated motion.

(b) the centre of mass of the body remains unchanged.

(c) the centre of mass of the body moves uniformly in a circular path

(d) individual particles and centre of mass of the body undergo an accelerated motion.

(J \& K CET 2008)

6. A $2 \mathrm{~kg}$ body and a $3 \mathrm{~kg}$ body are moving along the $x$-axis. At a particular instant the $2 \mathrm{~kg}$ body has a velocity of $3 \mathrm{~ms}^{-1}$ and the 3 $\mathrm{kg}$ body has the velocity of $2 \mathrm{~ms}^{-1}$. The velocity of the centre of mass at that instant is

(a) $5 \mathrm{~ms}^{-1}$

(b) $1 \mathrm{~ms}^{-1}$

(c) 0

(d) none of these

9. Two particles of equal mass have velocities $\overrightarrow{v_{1}}=2 \hat{i} \mathrm{~m} / \mathrm{s}$ and $\vec{v}_{2}=2 \hat{j} \mathrm{~m} / \mathrm{s}$. First particle has an acceleration $\overrightarrow{a_{1}}=(3 \hat{i}+3 \hat{j}) \mathrm{ms}^{-2}$ while the acceleration of the other particle is zero. The centre of mass of the two particles moves in a

(a) parabola

(b) circle

(c) straight line

(d) ellipse

12./A child is standing at one end of a long trolley moving with a speed $v$ on a smooth horizontal floor. If the child starts running towards the other end of the trolley with a speed $u$, the centre of mass of the system (child + trolley) will move with a speed

(a) zero

(b) $(v+u)$

(c) $(v-u)$

(d) $v$

15. A disc is rolling. The velocity of its centre of masses is $v_{\mathrm{cm}^{\prime}}$ Which one of the following statements is correct?

(a) Velocity of highest point and point of contact is $2 v_{\mathrm{cm}}$ each

(b) Velocity of highest point is $2 v_{\mathrm{cm}}$ and point of contact is zero

(c) Velocity of highest point is $2 v_{\mathrm{cm}}$ and that of point of contact is $v_{\mathrm{cm}}$

(d) Velocity of highest point is $v_{\mathrm{cm}}$ and that of point of contact is zero

(AIPMIT 2001)

16. A solid sphere of radius $R$ is placed on a smooth horizontal surface. A horizontal force $F$ is applied at height $h$ from the lowest point. For maximum acceleration of centre of mass, which is correct ?

(a) $h=0$

(b) $h=R$

(c) $h=2 R$

(d) No relation between $h$ and

(AIPMT 2002

17. The motion of the centre of mass is the result of

(a) internal forces

(b) external forces

(c) repulsive forces

(d) attractive forces

6.4 LINEAR MOMENTUM OF A SYSTEM OE PARTICLES

1. A machine gun fires a bullet of mass $40 \mathrm{~g}$ with a velocity 1200 $\mathrm{ms}^{-1}$. The man holding it can exert a maximum force of $144 \mathrm{~N}$ on the gun. How many bullets per sec. can be fire at the most ?

(a) one

(b) two

(c) three

(d) four

(Kerala PMT 2007)

2. A gun of mass $10 \mathrm{~kg}$ fires 4 bullets per sec. The mass of each bullet is $20 \mathrm{~g}$ and velocily of bullet when it leaves the gun is 300 $\mathrm{m} / \mathrm{s}$. The force required to hold the gun while firing is

(a) $6 \mathrm{~N}$

(b) $8 \mathrm{~N}$

(c) $24 \mathrm{~N}$

(d) $240 \mathrm{~N}$ (Odisha JEE 2008)

3. If the resultant of all the external forces acting on a system of particles is zero, one can surely say that

(a) Linear momentum of the system does not change in time

(b) $K E$ of system does not change in time

(c) $P E$ of system does not change in time

(d) Angular momentum of the system does not change with time

4. A bullet of mass $10 \mathrm{~g}$ moving with a velocity of $300 \mathrm{~m} / \mathrm{s}$ hits a block of ice if mass $5 \mathrm{~kg}$ and drops dead. The velocity of ice block is

(a) $60 \mathrm{~cm} / \mathrm{s}$

(b) $50 \mathrm{~cm} / \mathrm{s}$

(c) $40 \mathrm{~m} / \mathrm{s}$

(d) $30 \mathrm{~m} / \mathrm{s}$ (Odisha JEE 2009)

5. A bullet of mass $10 \mathrm{~g}$ is fired from a gun of mass $1 \mathrm{~kg}$. If the recoil velocity is $5 \mathrm{~m} / \mathrm{s}$, the velocity of the muzzle is

(a) $0.05 \mathrm{~m} / \mathrm{s}$

(b) $5 \mathrm{~m} / \mathrm{s}$

(c) $50 \mathrm{~m} / \mathrm{s}$

(d) $500 \mathrm{~m} / \mathrm{s}$

(Odisha JEE 2002)

6. The average resisting force that must act on a $5 \mathrm{~kg}$ mass to reduce its speed from $65 \mathrm{~cm} / \mathrm{s}$ to $15 \mathrm{~cm} / \mathrm{s}$ in $0.2 \mathrm{~s}$ is

(a) $-12.5 \mathrm{~N}$

(b) $25 \mathrm{~N}$

(c) $50 \mathrm{~N}$

(d) $100 \mathrm{~N}$

(EAMCET 2000)

$6.5$ GROSS PRODUCT OR

VECTOR PRODUCT OF TWO VECTORS

I. $\vec{A}$ and $\vec{B}$ are two vectors and $\theta$ is the angle between them, if $|\vec{A} \times \vec{B}|=\sqrt{3}(\vec{A} \cdot \vec{B})$ the value of $\theta$ is

(a) $60^{\circ}$

(b) $45^{\circ}$

(c) $30^{\circ}$

(d) $90^{\circ}$

(AIPMT 2007)

2. For vectors $\vec{A}$ and $\vec{B}$ making an angle $\theta$ which one of the following relations is correct?

(a) $\vec{A} \times \vec{B}=\vec{B} \times \vec{A}$

(b) $\vec{A} \times \vec{B}=A B \sin \theta$

(c) $\vec{A} \times \vec{B}=A B \cos \theta$

(d) $\vec{A} \times \vec{B}=-\vec{B} \times \vec{A}$

(DCE 2009)

3. If $\vec{A} \times \vec{B}=\vec{C}$, then which of the following statements is wrong ?

(a) $\vec{C} \perp \vec{A}$

(b) $\vec{C} \perp \vec{B}$

(c) $\vec{C} \perp(\vec{A}+\vec{B})$

(d) $\vec{C} \perp(\vec{A} \times \vec{B})$

9. The radius vector of a point is $\vec{r}=(\hat{i}-2 \hat{j}+3 \hat{k}) \mathrm{m}$ and a force $\vec{F}=(4 \hat{i}+5 \hat{j})$ acts at that point. The moment of the force in $\mathrm{Nm}$ is

(a) $(-15 \hat{i}+12 \hat{j}+13 \hat{k})$

(b) $(15 \hat{i}-12 \hat{j}+13 \hat{k})$

(c) $(-15 \hat{i}-12 \hat{j}+13 \hat{k})$

(d) $(15 \hat{i}+12 \hat{j}+13 \hat{k})$

12. The area of a parallelogram represented by the vectors $\vec{A}=2 \hat{i}+3 \hat{j} \quad$ and $\vec{B}=\hat{i}+4 \hat{j}$ is

(a) 14 units

(b) $7.5$ units

(c) 5 units

(d) 10 units

(Kerala PMT 200

13. If for two vectors $\vec{A}$ and $\vec{B} ; \vec{A} \times \vec{B}=0$, the vectors are

(a) perpendicular to eachother

(b) parallel to eachother

(c) acting at an angle of $60^{\circ}$

(d) acting at an angle of $30^{\circ}$

15. The resultant of two vectors having magnitude 2 and 3 is 1 . What is their cross product ?

(a) 6

(b) 3

(c) 1

(d) 0

18. The area of the triangle formed by $(2 \hat{i}+\hat{j}-\hat{k})$ and $(\hat{i}+\hat{j}+\hat{k})$ in square units

(a) 3

(b) $2 \sqrt{3}$

(c) $2 \sqrt{14}$

(d) $\frac{\sqrt{14}}{2}$

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