CHAPTER 6- SYSTEM OF PARTICLES AND ROTATIONAL MOTION

• Rigid body-
  • A hard body which does not affected by any external force.
  • And when we applied any external force, then all particle move same velocity in same direction, the distance between particles is same.
  • Flexible object is not rigid body.
• Translational motion-
  • The motion in which any rigid body move without changing its shape and all particles move with constant velocity in same direction.
  • If we slide the rectangle box form inclined plane, then it show translational motion, but when we slide the soft circle or cylinder object form inclined plane, then it not show translational motion.
• Rotational motion-
  • In a rigid body, every particle move in circle that lies in plane perpendicular to the axis and center of axis.
  • In rotational motion, body move in a particular axis, where all particle(which make this body) is move circularly.
• Arbitrary motion- The motion which does not follow any particular axis or fixed pattern or rule. This type of motion move in different direction with variable speeds.
• The motion of rigid body is may be or not fixed or pivoted. It can be pure translation motion, pure circular motion or mixture of circular and translation motion.

MASS ‘s CENTRE-

• If we have n particles of masses m_1, m_2, m_n on the line of x-axis, then-
  • ∑m_ix_i / ∑m_i, where x is distance of particle form origin. ∑ (Greek letter, sigma), denotes summation.
  • ∑m_i = M , total mass.
• In vector form-
  • R = (∑m_ir_i) / M.
• We can treat a body as mass are continuously distributed.

MOTION OF CENTRE OF MASS-

• The center of mass is the point where the total mass of a system can be considered.
• In motion, the center of mass move in the same way as if all the external forces acted on a single point.
• The motion of center of mass follow the Newton ‘s Law of Motion (constant velocity it no force acts, acceleration if force acts).
• Velocity of center of mass(constant), V= dR/dt.
• The total mass of particle times the acceleration of its center of mass is vector sum of all the forces acting on particles.
• The force of each particle will be external force exerted by body and internal force exerted by particle.
  • M A = F_ext, where F_ext is represent all external forces which acting on a particle.
• The center of mass of particle in system is moves as if all mass of system is was concentrated at a center of mass and at a point where all external forces were applied.

LINEAR MOMENTUM OF SYSTEM OF PARTICLE-

• The total momentum of particle in a system is equal to the product of total mass of system and velocity of mass of center.
  • P = M V
  • Suppose the sum of external force which acting on a system is zero.
• Conservation of linear momentum- It state that the total momentum of a closed system remains constant if no any external force acts on it.

PRODUCT OF VECTOR-

• The work is also define as a scalar product of two vector, force and displacement.
• Vector Product-
  • It is the product of 2 vectors a and b is vector c, where
    • c = ab sinθ, where a and b are magnitude of vector a and b,  θ is angle between 2 vectors.
    • Also c is perpendicular of vector a and b in plane.
• Properties of vector product-
  • a * b = -b * a
  • a * (b+c) = a*b + a*c
  • c = a * b

RELATION BETWEEN ANGULAR VELCITY AND LINEAR VELOCITY-

• Angular velocity- It is the rate at which an object rotates. It is measured as the angle turned per unit time.
• The role of angular velocity in rotational motion is, it determine how fast an object rotates and helps to calculate other quantities like angular displacement and angular acceleration.
• Instantaneous angular velocity is the velocity is an instant or velocity at point of time. It is denoted by ω.
• The ω is angular velocity of whole body.
• Relationship between angular velocity and linear-
  • v = r * ω, where v is velocity, r is radius (distance form the center of rotation), ω is angular velocity.
  • This means the linear velocity is product of radius and angular velocity.
• In rotation of fixed axis, the direction of ω is not change with time. But its magnitude may change form instant to instant. Generally, both magnitude and direction of ω may change form instant to instant.

ANGULAR ACCELERATION OF A BODY-

• Angular acceleration(α)- It is the rate of change of angular velocity with time.
• α = d ω/ dt, where dω is change in velocity and dt is change in time.

TORQUE(τ)-

• It is define as a vector product of 2 vectors.
• Torque(moment of force)-
  • Torque is the turning effect of a force about a pivot or axis.
  • Torque depends on the force applied and its perpendicular distance from the axis rotation
  • τ = r * F sinθ, where τ is torque, r is perpendicular distance form the axis to force line of action, F is applied force and θ is angle between r and F.

ANGULAR MOMENTUM(l)-

• It is define as a vector of 2 vector product.
• l = r * p, where l is angular momentum, p is position r relative to origin O.
• Angular momentum vector-
  • l = r p sinθ, where θ is angle between r and p.
• The relation between angular momentum and torque is-
  • τ = dl / dt.
  • That means torque(τ) is the rate at which angular momentum changes. If no external torque(τ) acts, angular momentum remains conserved.

ANGULAR MOMENTUM AND TORQUE FOR A SYSTEM OF PARTICLE-

• The rate of change of total angular momentum of a particle in a system is equal to sum of external torques is acting on a point in a system.
• In a system of a particle-
  • Torque(τ)- It is the total external torques acting on a system, calculated as the sum of torques on all particles
    • τ_ext = dl / dt
  • Angular Momentum(l)= It is the vector some of angular momenta of all particles in a system.

CONSERVATION OF ANGULAR MOMENTUM(l)-

• The conservation of angular momentum states that if no external torque acts on a system, then its total angular momentum remains constant.
• It means 3 component is conserved.

EQUILIBRIUM OF RIGID BODY-

Equilibrium of a Rigid Body

• Rigid Body Dynamics: We study how forces and torques affect rigid bodies (objects with a fixed shape).
  • Forces: Change the linear momentum of the rigid body.
  • Torques: Change the angular momentum of the rigid body.

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Mechanical Equilibrium

A rigid body is in mechanical equilibrium when:

1. Its linear momentum does not change (no linear acceleration).
2. Its angular momentum does not change (no angular acceleration).

This implies two key conditions:

1. Translational Equilibrium:
   Total force acting on the body is zero: ∑F_i ​= 0
   • (Vector sum of all forces = 0).
2. Rotational Equilibrium:
   Total torque acting on the body is zero:∑T_i ​= 0
   • (Vector sum of all torques = 0).

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Properties of Forces and Torques

• Origin Independence: Rotational equilibrium is independent of where you calculate torque if translational equilibrium holds.
• Components of Equations: Both conditions expand into 3 scalar equations for each axis (x, y, z), giving six independent conditions.

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Practical Scenarios

1. Coplanar Forces:
   • If forces act in one plane, only three conditions are needed:
     1. ∑F_x​=0 (Force balance in x-direction).
     2. ∑F_y​=0 (Force balance in y-direction).
     3. ∑T_z​=0 (Torque balance perpendicular to the plane).
2. Partial Equilibrium: A body can have:
   • Translational equilibrium only (forces balance, but torques don’t).
   • Rotational equilibrium only (torques balance, but forces don’t).

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Key Concepts

1. Couple:
   • Two equal and opposite forces not along the same line form a couple.
   • A couple creates rotation without translation.
   • Example: Turning a bottle lid applies a couple.
2. Principle of Moments:
   • For rotational equilibrium of a lever:
     • Load arm × Load = Effort arm × Effort
   • Mechanical Advantage (MA):
     • MA= Load​/Effort = Effort arm​/Load arm
3. Center of Gravity (CG):
   • The point where the gravitational torque on a body is zero.
   • For small bodies in uniform gravity, the center of gravity = center of mass.

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Worked Examples

Example 1: Two Parallel Forces on a Rod

1. If forces are equal and parallel:
   • No rotation (moments cancel out).
   • No translation (forces add to zero).
2. If forces are equal but opposite:
   • Rotation occurs (net torque ≠ 0).
   • No translation (forces cancel out).

Example 2: Ladder Leaning on a Wall

• Given: A 3m ladder, weight = 20 kg, rests against a frictionless wall with the foot 1m from the wall.
• To Find: Reaction forces.
  • Steps:
    1. Use translational equilibrium to balance vertical and horizontal forces.
    2. Use rotational equilibrium (torques about the foot of the ladder) to solve unknowns.
  • Results:
    • Reaction force of the wall:F₁ ​= 34.6 N.
    • Reaction force of the floor: F₂ ​= 199.0 N.

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Important Formulae

1. Torque: τ= F * R
2. Rotational Equilibrium: ∑τ = 0.
3. Translational Equilibrium: ∑f = 0.
4. Principle of Moments: Effort arm × Effort = Load arm × Load.
5. Mechanical Advantage: MA= Effort arm​ / Load arm​.

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Applications

1. Levers (e.g., see-saw, balance scales).
2. Couples (e.g., bottle caps, wrenches).
3. CG Finding:
   • Balance irregular shapes (e.g., cardboard).
   • Suspend and trace verticals to locate CG.

Moment of Inertia

1. Analogy in Rotational and Linear Motion

• Rotational motion is studied similar to translational motion.
• The equivalent of mass in rotational motion is the moment of inertia (I).

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2. Kinetic Energy in Rotational Motion

• For a body rotating about a fixed axis, each particle moves in a circle.
• Velocity of a particle: v_i = r_iω, where r_i is the distance from the axis, ωis angular velocity.
• Kinetic energy of a particle: K_i​ = ½ m_iv²_i =  ½ m_iv²_iω²
• Total kinetic energy of the body: K= ½  ​Iω²
  • where I = ∑mir²_i is the moment of inertia.

3. Moment of Inertia (I)

• I depends on:
  • Distribution of mass.
  • Distance of mass from the rotation axis.
  • Position and orientation of the axis.
• Formula: I = ∑m_ir_i²
• Units: kgm².
• Dimensions: [M¹L²T⁰].

4. Importance of Moment of Inertia

• III resists changes in rotational motion, similar to how mass resists changes in linear motion.
• It is not constant like mass; it depends on shape, size, and axis of rotation.

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5. Applications: Flywheel

• Flywheels in machines (e.g., engines) have large III.
• They smooth out rotational speed, reducing jerky motions and improving efficiency.

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6. Radius of Gyration (k)

• Defined as the distance from the axis where the entire mass M of the body could be concentrated to give the same I: I = MK²
  • k is a geometric property depending on shape and axis.

7. Examples of Moment of Inertia for Shapes

Shape	Axis	Moment of Inertia
Thin circular ring	Perpendicular at center	I=MR2
Thin rod	Perpendicular at midpoint	I = ML2​/12
Solid cylinder	Along axis	I = ½ MR2
Hollow cylinder	Along axis	I=MR2
Solid sphere	Diameter	I= 2/5 MR2

8. Kinematics of Rotational Motion

• Similar to linear motion, but angular quantities replace linear ones:
  • Angular displacement (θ) ↔ Displacement (x).
  • Angular velocity (ω) ↔ Velocity (v).
  • Angular acceleration (α) ↔ Acceleration (a).

Key Equations for Uniform Angular Acceleration:

• ω = ω₀ ​+ αt
• θ = θ₀ ​+ ω₀​t + ½ ​αt²
• ω² = ω²₀​ + 2α(θ−θ₀​)

9. Practical Problem (Example)

• A motor wheel increases speed from: ω₀= 1200 rpm to ω= 3120 rpmt= 16 seconds.
  • Convert ω to rad/s: ω= 2π × rpm/60.
  • Use α= (ω−ω₀)/t for angular acceleration.
  • Use θ= ω₀t + 1/2αt² to calculate angular displacement in radians.
  • Revolutions: Revolutions=θ/2π.

Dynamics of Rotational Motion About a Fixed Axis

1. Basics of Rotational Dynamics

• Rotational motion is analyzed similarly to linear motion, with corresponding quantities:
  • Displacement (x) ↔ Angular displacement (θ).
  • Velocity (v) ↔ Angular velocity (ω).
  • Acceleration (a) ↔ Angular acceleration (α).
  • Mass (m) ↔ Moment of inertia (I).
  • Force (F) ↔ Torque (τ).
• Work and energy relations also hold:
  • Work in linear motion: W=F⋅dx.
  • Work in rotational motion: W=τ⋅dθ.
• Rotational kinetic energy: K= 1/2 ​Iω².

2. Simplifications for Rotation About a Fixed Axis

• Consider only components of torque (τ) along the fixed axis; perpendicular components do not contribute.
• Forces and position vectors perpendicular to the axis are the primary contributors to torque.
• Torque (τ): τ= r * F * sin⁡(α), where α\alphaα is the angle between force and the radius vector.

3. Work Done by Torque

• Work done by a force in rotational motion: dW= τ * dθ.
• For multiple forces: Total torque (τ_total) is the algebraic sum of individual torques.
• Power in rotational motion: P= τ * ω, analogous to P= F * v in linear motion.

4. Newton’s Second Law for Rotation

• Torque produces angular acceleration: τ= Iα.
  • Moment of inertia (I) resists angular acceleration, similar to mass resisting linear acceleration.

5. Angular Momentum

• Angular momentum for a particle: L= r * p= m(r⋅v), where v=ωr⊥​.
• For a rigid body:
  • Angular momentum about the fixed axis: L_z​= Iω.
  • L_z​ is along the rotation axis for symmetric bodies.

6. Conservation of Angular Momentum

• If no external torque acts, angular momentum (L) remains constant: Iω=constant.
• Examples:
  • Rotating chairs: Extending arms reduces angular speed (ω) to conserve L.
  • Acrobatics: Adjusting body shape alters I, changing ω.

7. Key Equations

• Work: W= τ⋅θ.
• Power: P= τ⋅ω.
• Torque and angular acceleration: τ= Iα.
• Rotational kinetic energy: K= 1/2 ​Iω²..
• Angular momentum: L=Iω.

8. Summary Table: Translational vs. Rotational Motion

Linear Motion	Rotational Motion
Displacement: x	Angular displacement: θ
Velocity: v= dx/dt	Angular velocity: ω= dθ/dt
Acceleration: a= dv/dt	Angular acceleration: α= dω/dt
Mass: m	Moment of inertia: I
Force: F= ma	Torque: τ= Iα
Work: W= F * dx	Work: W= τ⋅dθ
Kinetic energy: 1/2 ​mv2	Kinetic energy: ½ Iω2
Power: P= Fv	Power: P=τω
Momentum: p= mv	Angular momentum: L=Iω

Applications

• Conservation of angular momentum in everyday activities like figure skating or acrobatics.
• Real-world problem-solving: calculating torque, energy, and angular motion parameters.

These concepts form the foundation of analyzing rotational systems and solving competitive questions on rotational dynamics.

These all are the notes of chapter 6 in physics. And after some time you get important questions and NCERT solutions HERE. *#THANKS FOR VISITING, VISIT AGAIN#* 😊