fyc7 system of particles & rotational …fyc7_ system of particles & rotational...

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FYC7_ SYSTEM OF PARTICLES & ROTATIONAL MOTION_CONCEPTUAL MCQS_(116Q)

1. A 2 kg body and a 3 kg body are moving along the x-axis. At a particular instant the 2 kg body has

a velocity of 3 ms–1 and the 3 kg body has the velocity of 2 ms–1. The velocity of the centre of mass

at that instant is

(1) 5 ms–1 (2)1 ms–1 (3) 0 (4) 2.4 ms–1

2. 2 bodies of different masses of 2 kg and 4 kg are moving with velocities 20ms–1 and 10ms–1

towards each other due to mutual gravitational attraction. What is the velocity of their centre of

mass

(1) 5 ms–1 (2) 6 ms–1 (3) 8 ms–1 (4) Zero

3. Choose the correct statement about the centre of mass (CM) of a system of two particles

(1) The CM lies on the line joining the two particles midway between them

(2) The CM lies on the line joining them at a point whose distance from each particle is inversely

proportional to the mass of that particle

(3) The CM lies on the line joining them at a point whose distance from each particle is proportional

to the square of the mass of that particle

(4) The CM is on the line joining them at a point whose distance from each particle is proportional to

the mass of that particle

____________________________________________________________________________________

Key : 1. (4) 2. (4) 3. (2)

___________________________________________________________________________________

Explanations

1. v�� For the motion along same line in the same direction, v��

∴ v�� 2.4ms��

2. m� � 2kg,m� � 4kg, v� � 20ms��, v� � �10ms��

v�� !�� ! � 0ms��.

3. By definition about C.M, m�r� � m�r� , mr � constant ⇒ r ∝ 1m


4. Two bodies of mass 1 kg and 3 kg have position vectors *̂ , -.̂ , /0 and �1*̂ � -.̂ , /0 ,

respectively. The centre of mass of this system has a position vector

(1) �2ı̂ , 2k3 (2) �2ı̂ � ȷ̂ , k3 (3) 2ı̂ � ȷ̂ , k3 (4) �ı̂ , ȷ̂ , k3

5. Assertion : The centre of mass of a system of proton and electron, released from their

respective positions remains at rest.

Reason : The centre of mass remain at rest, if no external force is applied.

(1) If both assertion and reason are true and the reason is the correct explanation of the assertion,

(2) If both assertion and reason are true but reason is not the correct explanation of the assertion.

(3) If assertion is true but reason is false.

(4)If the assertion and reason both are false.

6. Assertion : When a body dropped from a height explodes in mid air, its centre of mass keeps

moving in vertically downward direction.

Reason : Explosion occur under internal forces only.





____________________________________________________________________________________

Key : 4. (2) 5. (1) 6. (1)

___________________________________________________________________________________

Explanations

4. r�� 5��5�� 67̂�8̂9:; ��6��7̂��8̂9:;�� ∴ r�� 2ı̂ � ȷ̂ , k3

5. Initially the electron and proton were at rest so their centre of mass will be at rest. When they move

towards each other under mutual attraction then velocity of centre of mass remains unaffected

because external force on the system is zero.

6. Explosion is due to internal forces. As no external force is involved, the vertical downward motion of

centre of mass is not affected.


7. The centre of mass of a body

(1) Lies always outside the body

(2) May lie within, outside, on the surface of the body

(3) Lies always inside the body

(4) Lies always on the surface of the body

8. Two spheres of masses 2M and M are initially at rest at a distance R apart. Due to mutual force of

attraction, they approach each other. When they are at separation <- , the acceleration of the

centre of mass of spheres would be

(1) 0 (2) g ms–2 (3) 3g ms–2 (4) 12g ms–2

9. A cart of mass M is tied by one end of a massless rope of length 10 m. The other end of the rope is

in the hands of a man of mass M. The entire system is on a smooth horizontal surface. The man is

at x=0 and the cart at x=10 m. If the man pulls the cart by the rope, the man and the cart will meet

at the point

(1) x = 0 (2) x = 5m (3) x = 10m (4) They will never meet

10. In rotational motion of a rigid body, all particles move with

(1) Same linear & angular velocity

(2) Same linear and different angular velocity

(3) With different linear velocities and same angular velocities

(4) With different linear velocities and different angular velocities

____________________________________________________________________________________

Key : 7. (2) 8. (1) 9. (2) 10. (3)

___________________________________________________________________________________

Explanations:

7. Depends on the distribution of mass in the body

8. Initial acceleration of the system is zero. So it will always remain zero because there is no external

force on the system.

9. In the absence of external force, position of centre of mass remain same therefore they will meet at

their centre of mass.

10. As the body is rigid, angular velocity of all particles will be same i.e. ω= constant

but, linear velocity will be different for different particles such that, v ∝ r


11. A wheel is rotating at 900 rpm about its axis. When the power is cut-off, it comes to rest in 1

minute. The angular retardation in rads–2 is

(1) >� (2)

>! (3) >? (4)

>@

12. A wheel has angular acceleration of 3 rads–2 and an initial angular speed of 2 rads–1. In a time of

2seconds it has rotated through an angle of _______rad

(1) 6 (2)10 (3) 12 (4) 4

13. A sphere of mass 10kg and radius 0.5m rotates about a tangent. The moment of inertia of the

solid sphere is _______

(1) 5kgm� (2) 2.7kgm� (3) 3.5kgm� (4) 4.5kgm�

14. The moment of inertia of a body does not depend upon

(1) The angular velocity of the body (2) The mass of the body

(3) The distribution of mass in the body (4) The axis of rotation of the body

15. A circular thin disc of mass 2 kg has a diameter 0.2 m. Its moment of inertia about an axis

passing through the edge and perpendicular to the plane of the disc is ___ kgm2

(1) 0.01 (2) 0.03 (3) 0.02 (4) 3

____________________________________________________________________________________

Key : 11. (1) 12. (2) 13. (3) 14. (1) 15. (2)

___________________________________________________________________________________

Explanations:

11. ωD � �> E �? rads�� 30πrads��,ωH � 0, t � 60s ⇒ α � KL�KMN � �� >? � � >� rads��

12. θ � ωDt , �� αt� � 2� 2� , �� 3� 2�� 10rad.

13. I � Q�MR� � Q� � 10 � 0.5�� 3.5kgm�

14. (1)

15. I � ��MR� � �� 2 � 0.1�� 0.03kgm�


16. The radius of gyration of a disc of mass 50gm and radius 2.5 cm, about an axis passing through its

centre of gravity and perpendicular to the plane, is

(1) 0.52cm (2) 1.76cm (3) 3.54cm (4) 6.54cm

17. Moment of inertia of a ring of mass m = 3 gm and radius r = 1cm about an axis passing through its

edge and parallel to its natural axis is

(1) 10gmcm� (2) 100gmcm� (3) 6gmcm� (4) 1gmcm�

18. Radius of gyration of uniform thin rod of length L about an axis passing normally through its

centre of mass is

(1) T√�� (2)

T�� (3) √12L (4) 12 L

19. ABC is a triangular plate of uniform thickness. The sides are in the ratio shown in the figure WXY, WYZ, WZXare the moments of inertia of the plate about AB, BC, CA respectively. Which one of

the following relations is correct

(1) ICA

is maximum (2) IAB

> IBC

(3) IBC

> IAB

(4) IAB

+ IBC

= ICA

____________________________________________________________________________________

Key : 16. (2) 17. (3) 18. (1) 19. (3)

___________________________________________________________________________________

Explanations:

16. I � MK� � ��MR� ⇒ K � \√� � �.�√� � 1.76cm

17. I � 2MR� � 2 � 3 � 1�� 6gcm�

18. I � MK� � ]T�� ⇒ k � T√��

19. Distance of distribution of mass is such that the radius of gyration (K) is most about BC and least

about AC. ∴ K^_ ` Ka^ ` Ka_ ⇒ I^_ ` Ia^ ` Ia_


20. Two rings of the same radius and mass are placed such that their centres are at a common point

and their planes are perpendicular to each other. The moment of inertia of the system about an

axis passing through the centre and perpendicular to the plane of one of the rings is (mass of

ring = m, radius of ring = r)

(1) ��mr� (2) mr� (3)

��mr� (4) 2mr�

21. The ratio of the radii of gyration of a circular disc about a tangential axis in the plane of the disc

and of a circular ring of the same radius about a tangential axis in the plane of the ring is

(1) 2:3 (2) 2 : 1 (3) √5 ∶ √6 (4) 1: √2

___________________________________________________________________________________

Key : 20. (3) 21. (3)

___________________________________________________________________________________

Explanations:

20.

I1 = M.I. of ring about the axis normal to plane and passing through centre = mr2

I2 = M.I of ring about its diameter =

��mr�

For the system of two rings, IdedNf� � I� , I� � mr� , ��mr� � ��mr�

21. M.I. of disc about a tangent in its plane = �! mr� ∴ KgDd� � h�! r

M.I of ring about a tangent in its plane = ��mr� ∴ K5Dij � h�� r

∴ KgDd�K5Dij � h54

h32� k56


22. Out of the given bodies (of same mass) for which the moment of inertia will be maximum about

the axis passing through its centre of gravity and perpendicular to its plane

(1) Disc of radius a (2) Ring of radius a

(3) Square lamina of side 2a (4) Four rods each of length 2a making a square

23. Let l� be the force acting on a particle having position vector m�. If n��be the torque of this force

about the origin, then

(1) r�. τ�� 0andF��. τ�� 0 (2) r�. τ�� 0andF��. τ�� q 0

(3) r�. τ�� q 0andF��. τ�� 0 (4) r�. τ�� q 0andF��. τ�� q 0

24. If the torque acting upon a system is zero, then which of the following will be constant

(1) Force (2) Linear momentum

(3) Angular momentum (4) Linear impulse

____________________________________________________________________________________

Key : 22. (4) 23. (1) 24. (3)

__________________________________________________________________________________

Explanations:

22. M.I of disc = I� � ��ma�

M.I of ring = I� � ma�

M.I of square lamina of side 2a � I� � ]�� r 2a�� , 2a��s � ��Ma�

M.I of square system of side length 2a = I! � 4 tM �u�� ,Mv�u� w�x � �?� Ma�

Here, moment of inertia is maximum for square system of four rods.

23. τ�� r� � F�� ∴ τ��is z tobothr�andF�� So, r�. τ�� 0andF��. τ�� 0

24. τ � gTgN If τ � 0 then L = constant


25. A constant torque of 1000 Nm, turns a wheel of moment of inertia 200 kgm2 about an axis

through the centre. Angular velocity of the wheel after 3 s will be

(1)15rads�� (2) 10rads�� (3) 5rads�� (4) 1rads��

26. The instantaneous angular position of a point on a rotating wheel is given by the equation } � -~1 � �~-. The torque on the wheel becomes zero at

(1) t = 2s (2) t = 1s (3) t = 0.2s (4) t = 0.25s

27. A thin circular disc of mass M and radius R is rotating in a horizontal plane about an axis passing

through its centre and perpendicular to its plane with an angular velocity �. If another disc of

same dimensions but of mass �� is placed gently on the first disc co-axially, then the new angular

velocity of the system is

(1) �!ω (2)

��ω (3) !�ω (4)

��ω

__________________________________________________________________________________

Key : 25. (1) 26. (2) 27. (3)

__________________________________________________________________________________

Explanations:

25. Step 1 : α � �� 5rads��

Step 2 : ωH � ωD , αt ∴ ωH � 0 , 5� 30� � 15rads��

26. Torque zero means, the angular acceleration α� is zero.

Here, θ � 2t� � 6t� ∴ ω � g�gN � 6t� � 12t,α � gKgN � 12t � 12

For the given condition, α � 0 ∴ 12t � 12 � 0 ⇒t = 1 second.

27. By conservation of angular momentum

I�ω� � I�ω� ∴ ��MR�ω � �� MR� , �� v]!wR��ω� ⇒ ω� � !�ω


28. What remains constant in the field of central force

(1) Potential energy (2) Kinetic energy

(3) Angular momentum (4) Linear momentum

29. Two bodies have their moments of inertia I and 2I respectively about their axis of

rotation. If their kinetic energies of rotation are equal, then their angular momenta will be in the

ratio

(1) 1:2 (2)√2: 1 (3) 2 : 1 (4) 1 ∶ √2

30. Before jumping in water from above a swimmer bends his body to

(1) Increase moment of inertia (2) Decrease moment of inertia

(3) Decrease the angular momentum (4) Reduce the angular velocity

31. A ring of radius 0.5 m and mass 10 kg is rotating about its diameter with angular velocity of -�m��. Its kinetic energy is

(1) 10 J (2) 100 J (3) 500 J (4) 250 J

____________________________________________________________________________________

Key : 28. (3) 29. (4) 30. (2) 31. (4)

____________________________________________________________________________________

Explanations:

28. In the field central force, Torque = 0 ∴ angular momentum remains constant.

29. E � T�� ∴ L � � 2I�E

If E is constant then, T�T � h�� h �� √�

30. In doing so moment of inertia is decreased and hence angular velocity is increased.

31. Rotational kinetic energy = �� Iω� � �� v��MR�wω� � �� 10 � 0.5�� 20�� 250J


32. Two discs of same thickness but of different radii are made of two different materials such that

their masses are same. The densities of the materials are in the ratio 1:3.The moments of inertia

of these discs about the respective axes passing through their centres and perpendicular to their

planes will be in the ratio

(1) 1 : 3 (2) 3 : 1 (3) 1 : 9 (4) 9 : 1

33. A thin circular ring of mass M and radius r is rotating about its axis with a constant angular

velocity �. Two objects each of mass m are attached gently to the opposite ends of a diameter of

the ring. The ring will now rotate with an angular velocity

(1) K ]��]�� (2)

K]]�� (3) K]]� (4)

K ]��]

34. A child is standing with folded hands at the centre of a platform rotating about its central axis.

The kinetic energy of the system is K. The child now stretches his arms so that the moment of

inertia of the system doubles. The kinetic energy of the system now is

(1) 2K (2) �� (3)

�! (4) 4K

____________________________________________________________________________________

Key : 32. (2) 33. (2) 34. (2)

____________________________________________________________________________________

Explanations:

32. M.I. of disc = ��MR� � ��Mv ]>N�w � �� ]

>N� r∵ M � πR�tρs

If mass and thickness are same then, I ∝ �� ∴ ��

33. Initial angular momentum of ring = L � Iω � MR�ω

Final angular momentum of system (Ring + Two particles) = MR� , 2mR��ω′ As there is no external torque on the system, angular momentum is constant

∴ MR�ω � MR� , 2mR��ω� ∴ ω� � ]K ]��

34. KineticEnergy � K � T�� Here L = constant ∴ K ∝ �� ∴ �� ⇒ K� � ��


35. Assertion: If earth shrink (without change in mass) to half its present size, length of the day would

become 6 hours.

Reason: As size of the earth changes its moment inertia changes but angular momentum remains

same.





36. Assertion: Torque is equal to rate of change of angular momentum.

Reason: Angular momentum depends on moment of inertia and angular velocity.





____________________________________________________________________________________

Key : 35. (1) 36. (2)

____________________________________________________________________________________

Explanations:

35. When earth shrinks it angular momentum remains constant.

i.e. L � Iω � v�]\� wv�>� w � constant Here, M = constant ∴ T ∝ R� and

�� \\� � v�� w\� � �!

∴ T� � ��! � �!��5d! � 6hours

36. τ�� gT��gN andL � Iω


37. A flywheel gains a speed of 540 rpm. in 6 seconds. Its angular acceleration will be

(1) 3π rads–2 (2) 9π rads–2 (3) 90π rads–2 (4) 3π rads–2

38. A wheel rotates with a constant acceleration of 2 rads-2

. If the wheel starts from rest the number

of revolutions it makes in the first ten seconds will be approximately

(1) 8 (2)16 (3) 24 (4) 32

39. The direction of the angular velocity vector is along

(1) The tangent to the circular path (2) The inward radius

(3) The outward radius (4) The axis of rotation

40. A circular disc is to be made by using iron and aluminium, so that it acquires maximum moment of

inertia about its geometrical axis. It is possible with

(1) Iron and aluminium layers in alternate order

(2) Aluminium at interior and iron surrounding it

(3) Iron at interior and aluminium surrounding it

(4) Iron and aluminium layers in alternate order with alternate thickness

___________________________________________________________________________________

Key : 37. (1) 38. (2) 39. (4) 40. (2)

___________________________________________________________________________________

Explanations:

37. α � KN � �>v� ¡¢¡ w? � 3πrads��

38. Step 1

θ � ωDt , �� αt� ⇒ θ � 100rad Step 2

Number of revolution = x � ��> = � �> � 16(approximately)

39. Angular velocity is a axial vector and it is along the axis of rotation.

40. By doing so the distribution of mass is made away from the axis of rotation and hence, ‘K’ will be

large.


41. Three rings each of mass M and radius R are arranged as shown in the figure. The moment of

inertia of the system about YY’ will be

(1) 3MR� (2) ��MR� (3) 5MR� (4)

Q�MR�

42. A cylinder is of mass 500 gm and radius 10 cm. Its moment of inertia about its generator axis is

(1) 2.5 × 10–3 kgm2 (2) 2 × 10–3 kgm2 (3) 5 × 10–3 kgm2 (4) 3.5 × 10–3 kgm2

____________________________________________________________________________________

Key : 41. (4) 42. (1)

____________________________________________________________________________________

Explanations:

41. (4)

Moment of inertia of system about YY’ is I where,I � I� , I� , I�

Here ,I � I� , I� , I� = ��MR� , �� MR� , ��MR� � Q� MR�

42. (1)

I � ��MR� �� 0.5� 0.1�� 2.5 � 10��kgm�.


43. Three point masses m1, m

2, m

3 are located at the vertices of an equilateral triangle of length ‘a’.

The moment of inertia of the system about an axis along the altitude of the triangle passing

through m1, is

(1) m� ,m�� u! (2) m� ,m� ,m��a�

(3) m� ,m�� u! (4) m� ,m��a�

44. Of two eggs which have identical sizes, shapes and masses, one is raw and the other is half-boiled.

The ratio between the moments of inertia of the raw egg and that of the half- boiled egg about a

central axis is

(1) one (2) greater than one

(3) less than one (4) much less than one

___________________________________________________________________________________

Key : 43. (1) 44. (2)

___________________________________________________________________________________

Explanations:

43.

M.I of system about the axis which passing through m� is,

IdedNf� � m� 0�� ,m� vu�w� ,m� vu�w� ∴ IdedNf� � m� ,m�� u! 44. I5u¤fjj ` I�u¥H¦�D¥fgfjj. In boiled egg, mass is uniformly distributed. While in raw egg, mass is thrown away from the axis

due to centrifugal force.


45. The moment of inertia of uniform rectangular plate about an axis passing through its centre and

parallel to its length is (b = breadth of rectangular plate)

(1) ]¦! (2)]¦§? (3)

]¦§�� (4) ]¦��

46. One solid sphere A and another hollow sphere B are of same mass and same outer radii. Their

moments of inertia about their diameters are respectively IA and I

B such that

(1) IA � IB (2) Ia ` I^ (3) Ia ª I^ (4) Ia ≫ I^

47. Two rods each of mass m and length l are joined at the centre to form a cross. The moment of

inertia of this cross about an axis passing through the common centre of the rods and

perpendicular to the plane formed by them, is

(1) �¥�� (2)

�¥? (3) �¥§� (4)

�¥�

____________________________________________________________________________________

Key : 45. (4) 46. (3) 47. (2)

___________________________________________________________________________________

Explanations:

45.

M.I. of plate about O and parallel to length = ]¦��

46. Ia � Id�¥Dg � ��MR� � 0.4MR�,I^ � I¬�¥¥�¤ � �� MR� � 0.66MR� ∴ Ia ª I^.

47.

Moment of inertia of system about Z-axis = Ia^ , I_ � �¥�� , �¥�� ¥?


48. A uniform rod of length ‘2L’ has mass per unit length ‘m’. The moment of inertia of the rod about

an axis passing through its centre and perpendicular to its length is

(1) ��mL� (2)

��mL� (3) ��mL� (4)

!�mL�

49. A tap can be operated easily using two fingers because

(1) The force available for the operation will be more

(2) This helps application of angular forces

(3) A rotational effect is caused by the couple.

(4) The force by one finger overcomes friction and other finger provides the force for the operation

50. A constant torque acting on a uniform circular wheel changes its angular momentum from

A0 to 4A

0 in 4 seconds. The magnitude of this torque is

(1) �a¡! (2) A (3) 4A (4) 12A

51. A door 1.6 m wide requires a force of 1N to be applied at the free end to open or close it. The

force that is required at a point 0.4 m distant from the hinges for opening or closing the same

door is

(1)1.2 N (2) 3.6 N (3) 2.4 N (4) 4N

____________________________________________________________________________________

Key : 48. (3) 49. (3) 50.(1) 51. (4)

___________________________________________________________________________________

Explanations:

48. I � ]udd� TfijN�� T�� T�� T�!T�� T§�

49. (3)

50. τ � gTgN � T�T�∆N � !a¡�a¡! �a¡!

51. τ � constant ∴ r�F� � r�F� ∴ 16� 1� � 0.4� F�� ∴ F� � 4N

Contd.----------------


108. Assertion : It is harder to open and shut the door if we apply force near the hinge.

Reason : Torque is maximum at hinge of the door.

(1) If both assertion and reason are true and the reason is the correct explanation of the assertion.



(4) If the assertion and reason both are false.

109. A particle of mass ‘m’ is rotating in a plane in circular path of radius ‘r’. Its angular momentum is

‘L’. The centripetal force acting on the particle is

(1) T�5 (2)

T�5 (3) T�5 (4)

T�5§

110. The motion of planets in the solar system is an example of the conservation of

(1) mass (2) linear momentum

(3) angular momentum (4) energy

_______________________________________________________________________________________

Key : 108. 3 109. 4 110. 3

______________________________________________________________________________________

Explanations:

108. (3)

Torque = Tendency of rotation. It is the cross product of force and perpendicular distance of line of

action of force from the axis of rotation 6τ�� r� � F��;.

Hence, for a given applied force, torque of rotation will be high for large value of r.

109. (4)

Step 1: L � mvr ∴ v � T�5 Step 2: Centripetal force = F � ��5 � �5 v T�5w � T�5§

110. (3)

According to Kepler’s second law gagN � T��

Here m = mass of planet = constant and gagN = arial velocity of line which joins sun and the planet,

also a constant. Hence angular momentum (L ) is constant.


111. A circular turn table has a block of ice placed at its centre. The system rotates with an angular

speed � about an axis passing through the centre of the table. If the ice melts on its own without

any evaporation, the speed of rotation of the system

(1) becomes zero (2) remains constant at the same value ω

(3) increases to a value greater than ω (4) decreases to a value less thanω

112. A coin of radius r rolls without slipping on horizontal floor. If velocity of its centre of mass is v, the linear

velocity of point P on the coin is

(1) �� (2)

�� (3) 2v (4) ��

_______________________________________________________________________________________

Key : 111. 4 112. 1

_______________________________________________________________________________________

Explanations :

111. (4)

Melting of ice produces water which will spread over larger distance away from the axis of rotation.

This increases the moment of inertia of system. In the context L = Iω = constant, as I increases the

value of angular velocity ω� decreases.

112. (1)

Step 1 : v�� rω � v

Step 2 : Velocity of any point on the rolling body is the vector sum of the velocity of centre of mass and its

linear velocity

∴ v° � v�� , 5� ω� � v�� , 5K� � v�� , �±²� � ��±²� � ��

Contd.----------------

fyc7 system of particles & rotational …fyc7_ system of particles & rotational...

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