SYSTEM OF PARTICLES AND ROTATIONAL MOTION.pptx

This content is based on NEET Physics preparation for 11th students.

1. SYSTEM OF PARTICLES AND ROTATIONAL MOTION

2. SECTION 1

3. OVERVIEW System of particles and centre of mass Centre of mass of a system of particles - Centre of gravity and centre of mass Newton’s Law for a system of particles Motion of centre of mass Reduced Mass of a two particle system

4. SYSTEM OF PARTICLES Well-defined collection of a large number of particles that may or may not interact with each other or are connected to each other.

5. SYSTEM OF PARTICLES (Contd.) May be actual particles of rigid bodies in translational motion. The particle which interacts with each other apply force on each other These forces of mutual interaction between the particle of the system are called the internal force of the system.

6. SYSTEM OF PARTICLES (Contd.) Term External force means a force that is acting on any one particle, which is included in the system by some other body outside the system.

7. Centre of Mass The Centre of mass of a system of particles or a body is the point inside or outside the body where the entire mass of the body is assumed to be concentrated. Point at which algebraic sum of the mass moments of the particles reduces to zero.

8. Centre of mass The centre of mass may lie within, outside, on the surface of the body. Centre of mass at a point Centre of Mass for different shapes

9. Centre of mass

10. Centre of mass (Contd.) Centre of mass of system of particles is independent of origin Centre of Mass lies in the straight line joining two particles at location between the two particles Centre of mass of a two particle system, is located near the heavier particle

11. Centre of mass (Contd.)

12. Centre of mass (Contd.) If the origin is shifted to centre of mass i.e. to point P then The algebraic sum of moments of all masses of a system of particles, about the centre of mass, is zero. If the entire system net external force on the system is zero, the centre of mass remains unchanged.

13. Centre of Mass V s Centre of Gravity

14. Centre of Gravity and Centre of Mass If the gravitational field is uniform then the centre of mass and centre of gravity will coincide The term of Centre of gravity is meaningless where there is no gravity, while centre of mass is always valid The centre of gravity of a rigid body may change from place to place, while the centre of mass remains the same.

15. Newton’s Law for a System of Particles Newton's second law states that,  when a force acts on a particle, this force is equal to mass times acceleration .

16. Newton’s Law for a System of Particles Motion of Centre of Mass is governed by Newton’s law of motion The law is valid only for a closed system (No change in mass of the system)

17. Motion of centre of mass If a system experiences no external force, the center-of-mass of the system will remain at rest, or will move at constant velocity if it is already moving. The state of rest or uniform motion of the centre of mass of a body in a direction will not change unless it is acted upon by an external unbalanced force in that direction

18. Velocity, acceleration and linear Momentum of Centre of Mass

19. Block sliding on a frictionless wedge No external force in horizontal direction i.e. the X coordinate of the centre of mass will not change

20. TWO MASSES CONNECTED TO SPRING Two masses are stretched and the masses are released X 1 is the distance travelled by m 1 ( to right) X 2 is the distance travelled by m 2 ( to left)

21. Man walking on a Plank No external force in X direction, X coordinate of the centre of mass of the plank-man system will not change

22. EXPLOSION OF BOMB AT REST Mass of m explodes into three masses in m 1 , m 2 and m 3 Centre of mass continues to be at rest

23. Reduced Mass of a Two Particle System Consider a two particle system of masses m 1 and m 2 having velocities v 1 and v 2 , then velocity of centre of mass can be written as

24. Reduced Mass of a Two Particle System

25. SECTION 2

26. Overview Rigid body Centre of mass of rigid bodies Motion of a rigid body Rotation of a rigid body Axis rotation Moment of Inertia of a rigid body Parallel axes theorem Perpendicular axes theorem Moment of Inertia of some regular bodies

27. Rigid body Solid body in which the particles are arranged at definite positions so that the distance between the particle is fixed. Rigid bodies do not change the size and shape when external force is applied i.e. deformation is zero

28. Centre of mass of rigid bodies

29. Centre of mass of rigid bodies (Contd.)

30. Centre of mass of rigid bodies (Contd.)

31. Centre of mass of rigid bodies (Contd.)

32. Centre of mass of rigid bodies (Contd.)

33. Centre of mass of rigid bodies (Contd.)

34. Centre of mass of rigid bodies (Contd.)

35. Centre of mass – Continuous mass distribution Position vector of centre of mass Centre of mass (X direction) Centre of mass (Y direction) Centre of Mass (Z direction)

36. Motion of Rigid body Three types of Motion - Translatory motion - Combination of Translatory and Rotational motion - Pure Rotational motion

37. Translatory Motion During purely translational motion (motion with no rotation), all points on a rigid body move with the same velocity . Entire mass is assumed to be concentrates at the Centre of Mass System of forces acting can be reduced to single force passing through the Centre of Mass

38. Combination of Translation and Rotation System of forces acting on a body can be reduced to a single force not passing the centre of mass or two unequal parallel forces.

39. Rotation of a rigid body Fan rotating at a speed, with centre pivoted at a point Rigid body reduced to two equal unlike parallel forces

40. Rotation of a rigid body Pivot – applied force is equal and opposite to the force When the body rotates, the particles on it moves in circle, the centre of which lie on a straight line called “ Axis of rotation ” Body moves in a plane perpendicular to the axis of rotation. This plane is called “ plane of rotation ”

41. Important points on Translational and Rotational motion

42. Case studies

43. Moment of Inertia about a rigid body Property by virtue of which it opposes any change in its state of rest or uniform motion about an axis It may be compared to “mass” in translational motion Scalar quantity – Kg-m 2 Moment of Inertia is expressed as I = m × r 2 

44. Moment of Inertia – System of Particles The moment of inertia of a system of particles is given by, I = ∑ m i  r i 2   where r i  is the perpendicular distance from the axis to the ith particle which has mass m i .

45. Moment of Inertia (Contd.)

46. Parallel axis theorem The moment of inertia of a body about an axis parallel to the body passing through its centre is equal to the sum of the moment of inertia of the body about the parallel axis passing through the centre of mass and the product of the mass of the body times the square of the distance of between the two axes . Applied to body of any size and shape

47. Perpendicular axes theorem

48. Radius of Gyration

49. Radius of Gyration

50. Moment of Inertia of some regular bodies

51. Moment of Inertia of some regular bodies (Contd.)

52. Moment of Inertia of some regular bodies (Contd.)

53. SECTION 3

54. Overview Couple of forces Torque and angular acceleration Work Power Kinetic Energy of Rotation Work Energy Theorem Connected bodies Rolling

55. COUPLE OF FORCES Two equal and opposite forces with some finite perpendicular distance between them is called couple

56. COUPLE OF FORCES The perpendicular distance between the forces is called arm of couple Resultant force offered by a couple is zero Couple produces a torque Ꞇ is also called moment of force Torque or moment of force of couple about any point lying on the plane of the couple is the same

57. COUPLE OF FORCES

58. Torque Ability of a force to produce rotation in a body about an axis

59. Torque

60. Calculation of Torque

61. ANGULAR ACCELERATION - NEWTON’S LAW OF ROTATIONAL MOTION A body will continue in its state or rest or uniform rotational motion unless it is acted upon by an external unbalanced torque to change that state. The rate of change of angular momentum is equal to the net torque acting on the body Torque = Moment of inertia x angular acceleration

62. WORK

63. POWER

64. KINETIC ENERGY OF ROTATION

65. WORK ENERGY THEOREM

66. WORK ENERGY THEOREM

67. MOTION OF CONNECTED BODIES

68. MOTION OF CONNECTED BODIES

69. MOTION OF CONNECTED BODIES

70. MOTION OF CONNECTED BODIES

71. MOTION OF CONNECTED BODIES

72. MOTION OF CONNECTED BODIES

73. ROLLING Translational component V CM Tangential rotational component At point of contact P the net velocity is 0 Point O is having only translational velocity Point Q is the resultant of linear and angular component Point T has the linear and Tangential velocity adding up to the value 2V CM

74. KINETIC ENERGY IN ROLLING Kinetic energy of a body in rolling motion = (Kinetic Energy) translational + (Kinetic Energy) rotational

75. VELOCITY OF A ROLLING BODY

76. ACCELERATION OF A ROLLING BODY

77. SECTION 4

78. Overview Angular momentum of a point mass Angular momentum of a rigid body Relation between Angular Momentum (L) and Angular velocity Conservation of angular momentum Newton’s law of rotation motion Equilibrium of rigid bodies Relation between Angular Momentum (L) and kinetic Energy (KE) of a rotating body Angular Impulse Comparison of Linear and Rotational Motions

79. Angular momentum of a point mass

80. ANGULAR MOMENTUM OF RIGID BODY Angular momentum of a rigid body about an axis is the moment of linear momentum of the body about that axis

81. ANGULAR MOMENTUM OF RIGID BODY

82. ANGULAR MOMENTUM OF RIGID BODY

83. Relation between Angular Momentum (L) and Angular velocity

84. ANGULAR MOMENTUM OF RIGID BODY HAVING TRANSLATIONAL AND ROTATIONAL MOTION

85. ANGULAR MOMENTUM OF RIGID BODY

86. Conservation of Angular momentum Net external torque acting on a body or a system of particles is zero, the total angular momentum of the rigid body / system of particles is conserved.

87. Conservation of Angular momentum Examples Ballet dancer stretches arms - Moment of inertia - External torque is zero - Angular momentum is constant - Angular speed ( ω ) When she brings her arms towards her body ω increases

88. Conservation of Angular momentum Earth revolving around sun - Only force acting is the gravitational pull of sun on earth - This is the central force, so Ꞇ = 0 - Angular momentum remains constant

89. Conservation of Angular momentum Time (T 2 ) Angular velocity ( ω 2 )

90. NEWTON’S LAW OF ROTATIONAL MOTION A body will continue in its state or rest or uniform rotational motion unless it is acted upon by an external unbalanced torque to change that state. The rate of change of angular momentum is equal to the net torque acting on the body Torque = Moment of inertia x angular accleration

91. NEWTON’S LAW OF ROTATIONAL MOTION Resultant torque acting on a body about a point = moment of inertia of the body about that point x angular acceleration

92. Ꞇ = I α

93. Relation between Ꞇ and L

94. Equilibrium of rigid bodies

95. Angular Momentum (L) and Kinetic Energy (KE) We know that

96. ANGULAR IMPULSE

97. ANGULAR IMPULSE Product of torque acting on a body and the duration for which the torque acts is called angular impulse If the torque varies,

98. Comparison of Linear and Rotational Motion

99. Comparison of Linear and Rotational Motion (Contd.)

100. SECTION 5

101. OVERVIEW Rigid body in motion Pure translation Pure Rotation Rolling Pure and impure rolling Kinetic Energy in rolling Block sliding on a smooth Inclined plane Vs body rolling over a rough inclined plane Effect of external force on a body

102. RIGID BODY IN MOTION Pure translation - All particles move in straight line with same velocity - Acceleration of all particles are also the same in magnitude and direction Pure rotation - No translation - Angular velocity and angular acceleration are same

103. RIGID BODY IN MOTION (Contd.) Pure rotation - Linear velocity and acceleration of the particles may vary from point to point depending upon the distance from the axis of rotation - Velocity of any point = distance of the point from the axis x angular velocity

104. RIGID BODY IN MOTION (Contd.) Rolling - Combined motion of rotation and translation - Every point on the body will have two components of velocities - One component – translation velocity – same as the translational velocity of the centre of mass of the body - Second component – Velocity due to rotation of the body about its centre of mass

105. RIGID BODY IN MOTION (Contd.) Rolling

106. ROLLING

107. PURE ROLLING Adequate friction is present between the rolling surface and the body

108. IMPURE ROLLING (SLIDING) Inadequate friction (Relative velocity of the point of contact at the roller with the body >0) Velocity of

109. IMPURE ROLLING (SLIPPING) Inadequate friction (Relative velocity of the point of contact at the roller with the body <0 )

110. KINETIC ENERGY IN ROLLING Kinetic energy of a body in rolling motion = (Kinetic Energy) translational + (Kinetic Energy) rotational

111. Block sliding on a smooth Inclined plane Vs Body rolling over a rough inclined plane

112. Block sliding on a smooth Inclined plane Vs Body rolling over a rough inclined plane

113. Block sliding on a smooth Inclined plane Vs Body rolling over a rough inclined plane

114. Pure rolling Vs Impure rolling

115. Effect of External force on a body

116. Effect of External force on a body

117. Effect of External force on a body