MH12th| Physics| Chapter 1 Rotational Dynamics

Topics To Be Learn :

• Characteristics of circular motion
• Kinematics of circular motion
• Dynamics of circular motion
• Centripetal force and centrifugal force
• Applications of uniform circular motion
• Vehicle along a horizontal circular track
• Well of death
• Vehicle along a banked circular track
• Conical pendulum
• Vertical circular motion in Earth's gravity
• Point mass undergoing vertical circular motion in Earth's gravity
• Sphere (globe) of death
• Vehicle on a convex bridge
• Moment of a inertia (MI)
• Kinetic energy of a rotating body
• Ml of a uniform ring
• MI of a uniform disc
• Radius of gyration
• Theorems of parallel axis and perpendicular axes
• Angular momentum
• Expression for torque (in terms of MI)
• Conservation of angular momentum
• Rolling motion.

Circular Motion

• Definition: The motion of a particle along a complete circle or a part of it.

 Radius Vector:

• Definition: Particle's position vector with respect to the circle's center.
• Magnitude: Equals the circle's radius.
• Changes: Direction varies as the particle moves along the circumference.

Rotation and Revolution:

• Rotation: Circular motion about an axis passing through the body.
• Revolution: Circular motion around an axis outside the body.

Characteristics:

• Accelerated Motion: Velocity direction changes constantly.
• Periodic Motion: Particle repeats its path along the same trajectory.

Angular Displacement:

• Definition: Change in angular position of a particle with respect to a reference line passing through the circle's center.
• Magnitude: $𝛿𝜃={𝜃}_2-{𝜃}_1$.
• Direction: Perpendicular to the plane of revolution.

Angular Velocity:

• Definition: Time rate of angular displacement.
• Formula: $𝜔={\lim }_{𝛿𝑡\to 0}\frac{𝛿𝜃}{𝛿𝑡}$.
• Direction: Along the axis of rotation, determined by the right hand thumb rule.
• Also Called: Angular frequency, $𝜔=\frac{𝛿𝜃}{𝛿𝑡}$.

Right Hand Thumb Rule:

• Rule: If right hand fingers curl in the sense of particle revolution, thumb indicates angular displacement direction.

Linear Velocity in Circular Motion:

 

• Definition: Particle's velocity tangent to its circular path.
• Relation: $\mathbf{𝑣}=\mathbf{𝜔}\times \mathbf{𝑟}$
  • $\mathbf{𝑣}$: Linear velocity
  • $\mathbf{𝜔}$: Angular velocity
  • $\mathbf{𝑟}$: Radius vector
• Perpendicularity: $\mathbf{𝑣}$, $\mathbf{𝜔}$, and $\mathbf{𝑟}$ are mutually perpendicular.

Relation between Linear and Angular Velocity:

• Formula: $\mathbf{𝑣}=\mathbf{𝜔}\times \mathbf{𝑟}$
• Magnitudes: $𝑣=𝜔𝑟$ 

Uniform Circular Motion (UCM):

• Definition: Particle moves at constant linear speed or constant angular velocity in a circular path.
• Characteristic: Equal distances covered in equal time intervals.
• Periodic Motion: Repeats itself in equal time intervals.
• Examples:
  • Blades of a rotating fan.
  • Hands of a clock.
  • Earth-satellite in a circular orbit.

Period of Revolution (T):

• Definition: Time taken for one complete revolution.
• Formula: $𝑇=\frac{2𝜋𝑟}{𝑣}=\frac{2𝜋𝑟}{𝜔}$
  • $𝑟$: Radius
  • $𝑣$: Linear speed
  • $𝜔$: Angular speed

Frequency of Revolution (f):

• Definition: Number of revolutions per unit time.
• Formula: $𝑓=\frac{1}{𝑇}=\frac{𝑣}{2𝜋𝑟}=\frac{𝜔}{2𝜋𝑟}$
• SI Unit: Hertz (Hz)
• Dimensions: [M°L°T⁻¹]

Acceleration in Uniform Circular Motion (UCM):

• Definition: Acceleration of a particle moving at constant linear speed v in a circular path.
• Expression: $\overrightarrow{𝑎}=\frac{𝑑\overrightarrow{𝑣}}{𝑑𝑡}={𝑎}_{𝑟}\overrightarrow{𝑟}$
  • ${𝑎}_{𝑟}$: Radial acceleration (centripetal acceleration)
• Formula for Radial Acceleration: ${𝑎}_{𝑟}=-{𝜔}^2\overrightarrow{𝑟}$
  • $𝜔$: Angular velocity
• Magnitude of Radial Acceleration: $\mathrm{\mid }{𝑎}_{𝑟}\mathrm{\mid }={𝜔}^2𝑟=𝑣𝜔$
• Characteristics:
  • Perpendicular to the motion.
  • Constant in magnitude.
  • Directed radially inward, towards the centre of the circle.
  
  

Uniform Quantities in UCM:

• Linear speed $(𝑣)$ and angular speed $(𝜔)$.
• Also, kinetic energy, angular speed, and angular momentum.

Nonuniform Quantities in UCM:

• Velocity, acceleration, and centripetal force.
• Nonuniform Circular Motion: Particle's angular and linear speeds change with time.

Angular Acceleration ($𝛼$):

• Definition: Change in angular velocity over time.
• Formula: $\overrightarrow{𝛼}=\frac{𝑑\overrightarrow{𝜔}}{𝑑𝑡}=\frac{𝑑𝜔}{𝑑𝑡}$
• Magnitude of Average Angular Acceleration: $𝛼=\frac{{𝜔}_2-{𝜔}_1}{𝑡}$

 

 Tangential Acceleration in Nonuniform Circular Motion:

• Definition: Linear acceleration tangential to the path, causing a change in linear speed.
• Explanation:
  • If a particle in circular motion speeds up or slows down, both its linear speed $𝑣$ and angular speed $𝜔$ change.
  • Tangential acceleration $\overrightarrow{{𝑎}_{𝑡}}$ is along $\overrightarrow{𝑣}$.
  • Magnitude: ${𝑎}_{𝑡}=\frac{𝑑𝑣}{𝑑𝑡}$
  • If $𝑣$ increases, $\overrightarrow{{𝑎}_{𝑡}}$ is in the direction of $\overrightarrow{𝑣}$; if $𝑣$ decreases, $\overrightarrow{{𝑎}_{𝑡}}$ is opposite to $\overrightarrow{𝑣}$.

Relation between Linear and Angular Acceleration:

• Formula: ${𝑎}_{𝑡}=𝛼𝑟$
  • $𝛼$: Angular acceleration
  • $𝑟$: Radius of the circle

Resultant Acceleration in Nonuniform Circular Motion:

• Expression: $\overrightarrow{𝑎}=\overrightarrow{{𝑎}_{𝑡}}+\overrightarrow{{𝑎}_{𝑟}}$
  • $\overrightarrow{{𝑎}_{𝑟}}$: Radial or centripetal acceleration
  • Magnitude of $\overrightarrow{{𝑎}_{𝑟}}$: ${𝑎}_{𝑟}=𝜔𝑣$
  • Magnitude of $\overrightarrow{𝑎}$: $𝑎=\sqrt{{𝑎}_{𝑡}^2+{𝑎}_{𝑟}^2}$
• The angle between $\overrightarrow{{𝑎}_{𝑡}}$ and $\overrightarrow{{𝑎}_{𝑟}}$ is 90°.

Differences between Nonuniform and Uniform Circular Motion:

Nonuniform Circular Motion:

1. Angular and Tangential Accelerations:

   • Non-zero angular ($𝛼$) and tangential (${𝑎}_{𝑡}$) accelerations.
   • $𝛼=\frac{𝑑𝜔}{𝑑𝑡}$ and ${𝑎}_{𝑡}=\frac{𝑑𝑣}{𝑑𝑡}=𝛼\times 𝑟$
   • $𝛼$ in the direction of $𝜔$ if speeding up, opposite if slowing down.
   • ${𝑎}_{𝑡}$ in the direction of $𝑣$ if speeding up, opposite if slowing down.

2. Net Linear Acceleration:

   • Resultant of radial and tangential accelerations.
   • Not radially inward: $\overrightarrow{𝑎}=\overrightarrow{{𝑎}_{𝑡}}+\overrightarrow{{𝑎}_{𝑟}}$

3. Centripetal Acceleration and Force:

   • Magnitudes not constant.

Example: Motion of a fan blade when the fan speeds up or slows down.

Uniform Circular Motion:

1. Angular and Tangential Accelerations:

   • Zero angular and tangential accelerations.
   • Angular speed ($𝜔$) and linear speed ($𝑣$) are constant.

2. Net Linear Acceleration:

   • Radially inward: $\overrightarrow{𝑎}$ is centripetal.

3. Centripetal Acceleration and Force:

   • Magnitudes are constant.
   Example: Motion of clock hands.

Kinematical Equations for Circular Motion:

• $𝜔={𝜔}_0+𝛼𝑡$
• $𝜃-{𝜃}_0={𝜔}_{𝑎𝑣}\times 𝑡$
• ${𝜔}^2={𝜔}_0^2+2𝛼(𝜃-{𝜃}_0)$

Where:

• ${𝜔}_0$ and $𝜔$ are initial and final angular speeds.
• $𝑡$ is time, ${𝜃}_0$ and $𝜃$ are initial and final angular positions.
• $𝛼$ is the angular acceleration.
• ${𝜔}_{𝑎𝑣}$ is the average angular speed.

Centripetal Force:

• Definition:

  • Force pointing radially towards the center of the circle in uniform circular motion.
  • Produces centripetal acceleration keeping the particle in its circular path.

• Explanation:

  • Necessary for the particle to maintain its circular motion.
  • If absent, the particle would move tangentially away from the circle.
  • Its direction changes as the particle moves around the circle.

• Examples:

  • Gravitational force on an Earth-satellite.
  • Coulomb force in Bohr's atom.
  • Tension in a revolving string.
  • Force of static friction in a car turning on a road.

Centrifugal Force:

• Definition:

  • Fictitious, outward force acting on a particle in its reference frame in circular motion.
  • Equal in magnitude to the centripetal force but opposite in direction.

• Explanation:

  • Appears in a non-inertial frame due to the acceleration of the reference frame.
  • Fictitious force to explain the outward tendency in the particle's reference frame.

• Examples:

  • Person on a merry-go-round.
  • Passengers in a turning car.
  • Coin on a rotating turntable.
  • Earth's centrifugal force causing equatorial bulge.

 Difference between Centripetal Force and Centrifugal Force:

 

 Applications of Uniform Circular Motion:

Expression for Maximum Safe Speed on a Horizontal Circular Road:

• Expression: ${𝑣}_{\text{max}}^2=𝑟{𝜇}_{𝑠}𝑔$

• Significance:

  • The maximum safe speed ${𝑣}_{\text{max}}$ depends on the coefficient of static friction (μs), which varies with road conditions.
  • If friction is insufficient, the vehicle may skid off the road.
  • On a level railway track, the outward thrust on the outer rail due to the flange of outer wheels provides necessary centripetal force.
  • Risk of Rolling Outward: Due to friction, vehicles may tend to roll outward, especially on unbanked curves.
  • Bicyclist's Leaning Inward: To counteract outward torque, a bicyclist leans inward on unbanked roads.
  

Maximum Safe Speed for a Vehicle on Circular Road

• Concept:

  To prevent rollover, the maximum safe speed for a vehicle on a circular road must be determined.

• Forces on the Car:

  • Lateral Limiting Force of Static Friction (fs): Acts along the axis of the wheels towards the center of the circular path.
  • Weight (mg): Acts vertically downwards at the center of gravity (C.G.).
  • Normal Reaction (N):Acts vertically upwards at the C.G.

• Equations:

  • Centripetal Force = Limiting Force of Static Friction:
    • $𝑚{𝑣}^2\mathrm{/}𝑟=𝑓𝑠$
  • Torque Produced by Friction Force (τt):
    • $𝜏𝑡=𝑓𝑠\cdot ℎ=(𝑚{𝑣}^2\mathrm{/}𝑟)\cdot ℎ$
  • Torque Produced by Weight (τr):
    • $𝜏𝑟=𝑚𝑔\cdot (𝑤\mathrm{/}2)$
  • Condition for No Toppling:
    • $𝜏𝑡\ge 𝜏𝑟$
  • Maximum Safe Speed (v):
    • $(𝑚{𝑣}^2\mathrm{/}𝑟)\cdot ℎ=𝑚𝑔\cdot (𝑤\mathrm{/}2)$

• Key Points:

  • Roll: Rotation about the front-to-back axis.
  • Restoring Torque: Opposite torque that restores the car back on all four wheels.
  • Rollover: Occurs when the gravitational force passes through the pivot point of the outer wheels.
  • Static Stability Factor (SSF): Ratio of track width to height of the center of gravity.
  • Risk of Rollover:
    • Low if SSF ≤ ${𝜇}_{𝑠}$.
  • Skidding vs Rollover: Skidding more likely if
    • ${𝜇}_{𝑠}$ is low, such as on wet or icy roads.

• Equations Simplified:

  • $\frac{𝑓𝑠\cdot 𝑚𝑔}{{𝑤}^2}\ge 1$
  • ${𝜇}_{𝑠}\le \frac{{𝑤}^2}{ℎ}$

• Factors Affecting Maximum Safe Speed:

  • Larger Track Width: Increases maximum safe speed.
  • Lower Center of Gravity: Increases maximum safe speed.
  • Friction Coefficient (μs): Higher value decreases the risk of rollover.

Banking of a Road

• Definition:
  • Banking of Road: Tilting the road surface at a bend, raising the outer side above the inner side.
• Need for Banking:
  • Circular Motion: Car performs circular motion while turning.
  • Centripetal Force: On horizontal road, provided by static friction.
  • Friction: Depends on surface and road conditions, not reliable.
• Advantages of Banking:
  • Centripetal Force on Banked Road:
    • Resultant of normal reaction and gravitational force.
    • Eliminates dependence on friction.
  • Wear and Tear: Reduced, as friction is not relied upon.
• Expression for Optimum Speed on Banked Road:
  • Forces on Car:
    • Weight (mg), Normal Reaction (N), Frictional Force (fs).
  • At Optimum Speed:
    • Friction not relied upon.
    • Normal reaction (N) resolves into components:
      • $𝑁\cos 𝜃$ balances weight (mg).
      • $𝑁\sin 𝜃$ provides necessary centripetal force.
    • $𝑁\sin 𝜃=\frac{𝑚{𝑣}_{𝑜}^2}{𝑟}$
    • $𝑁\cos 𝜃=𝑚𝑔$
    • $\tan 𝜃=\frac{{𝑣}_{𝑜}^2}{𝑟𝑔}$
• Factors Affecting Optimum Speed:
  • Angle of Banking: Determines the tilt of the road.
  • Radius of Curved Path: Influences the optimum speed.
• Equation Simplified:
  • $𝜃={\tan }^{-1}\left(\frac{{𝑣}_{𝑜}^2}{𝑟𝑔}\right)$
  • ${𝑣}_{𝑜}=\sqrt{𝑟𝑔\tan 𝜃}$
• Key Points:
  • Independent of Vehicle Mass: Angle of banking.
  • Dependent on Road Conditions: Angle of banking and radius of the curved path.
   

Two Factors Affecting Safe Speed on Banked Road:

1. Angle of Banking of Road
2. Radius of Curved Path

Minimum and Maximum Safe Speed on Banked Road

• Concept: Minimum and maximum speeds for safe negotiation of a banked curve without skidding.
• Forces on Car: Weight (mg), Normal Reaction (N), Frictional Force (fs).
• Minimum Safe Speed:
  • Condition: Car driven slower than optimum speed (vo).
  • Components of Normal Reaction and Friction:
    • $𝑁\cos 𝜃$ and $𝑓𝑠\sin 𝜃$ vertically up.
    • $𝑁\sin 𝜃$ and $𝑓𝑠\cos 𝜃$ horizontally towards the center.
  • Equations:
    • ${𝑣}_{\text{min}}=\sqrt{\frac{𝑟𝑔(\tan 𝜃-{𝜇}_{𝑠})}{1+{𝜇}_{𝑠}\tan 𝜃}}$
    • ${𝜇}_{𝑠}\le \tan 𝜃$: ${𝑣}_{\text{min}}=0$
• Maximum Safe Speed:
  • Condition: Car driven faster than optimum speed (vo).
  • Components of Normal Reaction and Friction:
    • $𝑁\cos 𝜃$ vertically up.
    • $𝑓𝑠\sin 𝜃$ vertically down.
    • $𝑁\sin 𝜃$ and $𝑓𝑠\cos 𝜃$ horizontally towards the center.
  • Equations:
    • ${𝑣}_{\text{max}}=\sqrt{\frac{𝑟𝑔(\tan 𝜃+{𝜇}_{𝑠})}{1-{𝜇}_{𝑠}\tan 𝜃}}$
    • ${𝜇}_{𝑠}=\cot 𝜃$ for $𝜃\ge 45^{\circ }$: ${𝑣}_{\text{max}}=\mathrm{\infty }$
• Factors Affecting Safe Speed:
  • Angle of Banking (θ): Determines optimum and safe speeds.
  • Friction Coefficient (μs): Influences minimum and maximum speeds.
• Friction and Locomotion:
  • Necessary for Motion: Without friction, a vehicle cannot move.
  • Kinetic Friction: If car skids, friction becomes kinetic, changing direction abruptly.

Expression for Minimum and Maximum Safe Speeds on Banked Road:

• Minimum Safe Speed:
  • Formula: ${𝑣}_{\text{min}}=\sqrt{\frac{𝑟𝑔(\tan 𝜃-{𝜇}_{𝑠})}{1+{𝜇}_{𝑠}\tan 𝜃}}$
  • Condition: ${𝜇}_{𝑠}\le \tan 𝜃$
  • Special Case: ${𝑣}_{\text{min}}=0$ if ${𝜇}_{𝑠}\le \tan 𝜃$
• Maximum Safe Speed:
  • Formula: ${𝑣}_{\text{max}}=\sqrt{\frac{𝑟𝑔(\tan 𝜃+{𝜇}_{𝑠})}{1-{𝜇}_{𝑠}\tan 𝜃}}$
  • Special Case: ${𝑣}_{\text{max}}=\mathrm{\infty }$ if ${𝜇}_{𝑠}=\cot 𝜃$ for $𝜃\ge 45^{\circ }$

 

Factors Affecting Safe Speed:

• Angle of Banking (θ)
• Friction Coefficient (μs)

Conical Pendulum

• Definition: A simple pendulum where the bob moves in a horizontal circle, following the surface of an imaginary right circular cone.
• Expression for Angular Speed:
  • Conditions:
    • Bob performs Uniform Circular Motion (UCM) in a horizontal plane.
    • String makes a constant angle $𝜃$ with the vertical.
  • Forces on Bob:
    • Weight ($𝑚𝑔$) and Tension ($𝑇$).
  • Centripetal Force:
    • ${𝑇}_0=𝑚{𝜔}^2𝑟$
  • Resolving Tension:
    • ${𝑇}_0\cos 𝜃$ balances weight.
    • ${𝑇}_0\sin 𝜃$ provides centripetal force.
  • Equations:
    • ${𝑇}_0\sin 𝜃=𝑚{𝜔}^2𝑟$
    • ${𝑇}_0\cos 𝜃=𝑚𝑔$
    • $\tan 𝜃=\frac{{𝜔}^2𝑟}{𝑔ℎ}$
    • $\cos 𝜃=\frac{𝑔}{{𝜔}^2𝐿}$
    • $𝜔=\sqrt{\frac{𝑔}{𝐿\cos 𝜃}}$

• Key Points:
  • Angular Speed (ω):
    • Increases as $𝜃$ decreases.
    • Decreases as $𝜃$ increases.
  • Relationship with $𝜃$:
    • As $𝜔$ increases, $\cos 𝜃$ decreases, $𝜃$ increases.

Expression for Angular Speed of Conical Pendulum:

• $𝜔=\sqrt{\frac{𝑔}{𝐿\cos 𝜃}}=\sqrt{\frac{𝑔}{𝐿}}\sqrt{\frac{1}{\cos 𝜃}}$

Frequency of Revolution of Conical Pendulum

• Definition:

  • Frequency ($𝑛$): Number of revolutions of the bob in one second.

• Expression:

  • If $𝑛$ is the frequency of revolution of the bob:
    • $𝜔=2𝜋𝑛=\sqrt{\frac{𝑔}{𝐿\cos 𝜃}}$
    • $𝑛=\frac{1}{2𝜋}\sqrt{\frac{𝑔}{𝐿\cos 𝜃}}$

• Key Points:

  • $𝑛$ is proportional to $\sqrt{𝑔}$ (acceleration due to gravity).
  • $𝑛$ is proportional to $\frac{1}{\sqrt{𝐿}}$.
  • $𝑛$ is proportional to $\frac{1}{\sqrt{\cos 𝜃}}$ (as $𝜃$ increases, $\cos 𝜃$ decreases, $𝑛$ increases).
  • Frequency is independent of the mass of the bob.

Period of Conical Pendulum

• Definition:

  • Period (${𝑇}_{𝑃}$): Time taken by the bob to complete one revolution.

• Expression:

  • ${𝑇}_{𝑃}=\frac{2𝜋}{𝜔}=\frac{2𝜋}{\sqrt{\frac{𝑔}{𝐿\cos 𝜃}}}=2𝜋\sqrt{\frac{ℎ}{𝑔}}$
    • $ℎ$: Axial height of the cone.

• Key Points:

  • ${𝑇}_{𝑃}$ is proportional to $\frac{1}{\sqrt{𝐿}}$.
  • ${𝑇}_{𝑃}$ is proportional to $\sqrt{𝑔}$.
  • ${𝑇}_{𝑃}$ is proportional to $\sqrt{\cos 𝜃}$ (as $𝜃$ increases, $\cos 𝜃$ and ${𝑇}_{𝑃}$ decreases).
  • Period is independent of the mass of the bob.
    

 Vertical Circular Motion

• Definition: Body revolving in a vertical circle under Earth's gravitational field.
• Nonuniform Circular Motion: Linear speed not constant, but motion can be periodic.
• Controlled Vertical Circular Motion: Linear speed controlled, can be constant or zero.
• Minimum Speeds:
  • Nonuniform Motion:
    • Body attached to a string or in loop-the-loop maneuvers.
    • Minimum speed required to complete circle.
  • Controlled Motion:
    • Body attached to a rod or in Ferris wheel ride.
    • Zero speed possible at top.

 

Expressions for Minimum Speeds

• Using Energy Conservation:
  • For Particle Attached to String: Two forces: Weight $𝑚𝑔$ and Tension along string.
    
    • Minimum Speed Required: To prevent slackening of string before reaching top.
• Key Points:
  • Nonuniform Circular Motion: Minimum speed required throughout.
  • Controlled Circular Motion: Minimum speed required only at certain points.

Minimum Speeds in Vertical Circular Motion

• Top Position (Point A):
  • Let ${𝑣}_{𝐴}$ be the speed of the particle, ${𝑇}_{𝐴}$ the tension in the string.
  • Net force towards the center: ${𝑇}_{𝐴}+𝑚𝑔$
  • Equation:
    • ${𝑇}_{𝐴}+𝑚𝑔=\frac{𝑚{𝑣}_{𝐴}^2}{𝑟}$
  • Minimum Speed (Minimum Energy):
    • ${𝑇}_{𝐴}=0$
    • $𝑚𝑔=\frac{𝑚{𝑣}_{𝐴}^2}{𝑟}$
    • ${𝑣}_{𝐴}^2=𝑟𝑔$
    • ${𝑣}_{𝐴}=\sqrt{𝑟𝑔}$
• Bottom Position (Point B):
  • Let ${𝑣}_{𝐵}$ be the speed at bottom.
  • Total Energy at Bottom:
    • $𝐾𝐸+𝑃𝐸=\frac{1}{2}𝑚{𝑣}_{𝐵}^2+0=\frac{1}{2}𝑚{𝑣}_{𝐵}^2$
  • Total Energy at Top:
    • $𝑃𝐸=𝑚𝑔ℎ=𝑚𝑔(2𝑟)$
    • $𝐾{𝐸}_{\text{min}}=\frac{1}{2}𝑚{𝑣}_{𝐴}^2=\frac{1}{2}𝑚𝑔𝑟$
    • Equation:
      • $\frac{1}{2}𝑚{𝑣}_{𝐵}^2=\frac{5}{2}𝑚𝑔𝑟$
      • ${𝑣}_{𝐵}=\sqrt{5𝑔𝑟}$
• Midway (Point C):
  • Let ${𝑣}_{𝐶}$ be the speed at point C.
  • Total Energy at C:
    • $𝐾𝐸+𝑃𝐸=\frac{1}{2}𝑚{𝑣}_{𝐶}^2+𝑚𝑔𝑟$
  • Conservation of Energy:
    • $\frac{1}{2}𝑚{𝑣}_{𝐶}^2+𝑚𝑔𝑟=\frac{5}{2}𝑚𝑔𝑟$
    • ${𝑣}_{𝐶}^2=3𝑔𝑟$
    • ${𝑣}_{𝐶}=\sqrt{3𝑔𝑟}$

Minimum Speeds:

• At Top (Point A): ${𝑣}_{𝐴}=\sqrt{𝑟𝑔}$
• At Bottom (Point B): ${𝑣}_{𝐵}=\sqrt{5𝑔𝑟}$
• At Midway (Point C): ${𝑣}_{𝐶}=\sqrt{3𝑔𝑟}$

Vertical Circular Motion as Nonuniform Circular Motion

• Body in Vertical Circular Motion: Body of mass m tied to string, revolving in vertical circle of radius r.
• Forces Acting:
  • Weight mg and tension T in the string.
  • Resolving weight into components: mg cosθ (radial) and mg sinθ (tangential).
• At Point P:
  • Net force towards center: T - mg cosθ.
  • Centripetal force: T - mg cosθ = mv²/r.
• At Lowest Point B: Kinetic energy at B: ½mv_B² (Potential energy = 0).
• Total Energy at Point P:
  • Total energy = Kinetic energy + Potential energy.
  • Rise through height h = r(1 - cosθ).
  • Equation: Total energy at P = ½mv² + mgh = ½mv² + mgr(1 - cosθ).
• Conservation of Total Energy:
  • Total energy at any point = Total energy at bottom.
  • From Eqs. (2) and (3): ½mv² + mgr(1 - cosθ) = ½mv_B².
• Expression for Speed $𝑣$:
  • v² = v_B² - 2gr(1 - cosθ).
  • v = √(v_B² - 2gr(1 - cosθ)).
• Conclusion:
  • Linear speed $𝑣$ changes with θ.
  • As θ increases, v decreases.
  • As θ decreases, v increases.
  • Vertical circular motion controlled by gravity is nonuniform circular motion.

Expression for Tension in String

• Substituting for ${𝑣}^2$ from Eq. (4) into Eq. (1): T = mv²/r - mg(2 - 3cosθ).

Moment of Inertia (MI) and Kinetic Energy of a Rotating Body

Moment of Inertia (MI):

• Definition: Defined as the sum of products of masses of body particles and squares of their distances from the axis of rotation.
• For Discrete Particles:
  • If body has N particles with masses m1, m2, ..., mN at distances r1, r2, ..., rN:
    • I = m1r₁² + m2r₂² + ... + mNrN²
    • I = Σ(mirᵢ²)
     
• For Rigid Body:
  • With continuous and uniform mass distribution:
    • I = ∫r²dm
    • dm is mass of infinitesimal element at distance r from axis.
• Depends On:
  • (i) Mass and shape of body
  • (ii) Orientation and position of rotation axis
  • (iii) Distribution of mass about rotation axis
• Dimensions:
  • Moment of Inertia = Mass × (Distance)²
  • [Moment of Inertia] = [M] [L²] = [M1L²T⁰]
• SI Unit:
  • Kilogram-metre² (kg-m²).

Kinetic Energy of a Rotating Body:

 

• Kinetic Energy (KE) of Rotating Body:
  • Kinetic energy of rotating body depends on its moment of inertia and angular velocity.
• Expression for KE:
  • KE = ½Iω²
  • ω is angular velocity of rotation.
• Depends On:
  • Moment of inertia (I)
  • Angular velocity (ω).

Physical Significance of Moment of Inertia

• Linear Motion

  1. Momentum = mass × velocity
  2. Force = mass × acceleration
  3. Kinetic energy = ½ mv²

• Rotational Motion

  1. Angular momentum = moment of inertia × angular velocity
  2. Torque = moment of inertia × angular acceleration
  3. Kinetic energy = ½ Iω²

• Comparison:

  • Force ↔ Torque
  • Momentum ↔ Angular momentum
  • Mass ↔ Moment of inertia
  • Velocity ↔ Angular velocity
  • Acceleration ↔ Angular acceleration

• Physical Significance:

  • Moment of inertia in rotational motion is like mass in linear motion.
  • It's the rotational inertia, opposing changes in angular velocity.
  • In absence of net torque, body maintains uniform angular velocity.
    

Moment of Inertia (MI)

Thin Ring:

• Description:
  • Uniform mass distributed along circumference
  • Negligible radial thickness compared to radius
• Formula: Moment of Inertia (I) = MR²

Thin Uniform Disc:

• Description:
  • Uniform mass distributed over circular surface
  • Axial thickness negligible compared to radius
• Formula: Moment of Inertia (I) = ½ MR²

Radius of Gyration

• Definition:

  Distance between rotation axis and point where entire mass can be concentrated to give same moment of inertia.

• Formula:

  • Moment of Inertia (I) = Mk²
  • Radius of Gyration (k) = √(I/M)

• Physical Significance:

  • Indicates mass distribution around rotation axis.
  • Smaller k if mass closer to axis, larger k if mass away from axis.

• Variability:

  • Depends on mass distribution.
  • Changes with choice of rotation axis.
    

Theorem of Parallel Axis

• Definition:

  • Moment of inertia (MI) about an axis equals sum of:
    1. MI about parallel axis through center of mass.
    2. Product of mass and square of distance between axes.

• Proof:

  • MI about axis ACB: IC = ∫(DC)²dm
  • MI about axis MOP: IO = ∫(DO)²dm
  • IO = IC + Mh² (where h is distance between axes)

• Applicability:

  • Applicable to any body.
  • If MI about axis through center of mass is known (ICM), then:
    • I = ICM + Mh²

 

Theorem of Perpendicular Axes

• Definition:

  MI of a plane lamina about perpendicular axis equals sum of MI about two perpendicular axes in lamina's plane and through intersection of perpendicular axis and lamina.

• Proof:

  • MI about z-axis: IZ = ∫(OP)²dm
  • MI about x-axis: Ix = ∫y²dm
  • MI about y-axis: Iy = ∫x²dm
  • IZ = Ix + Iy
   

Moment of Inertia of Uniform Rod

• About Transverse Axis Through Centre:

  ICM = ML²/12

• About Transverse Axis Through One End:

  I = ML²/3

• Radii of Gyration:

  • k (Centre Axis) = L/√12
  • k (End Axis) = L/√3

Expression for Moment of Inertia of Thin Ring

• About Tangential Axis:

  ICM = MR²

• Radius of Gyration:

  k = R
  

Moment of Inertia (MI) of a Thin Ring

• About a Tangent Perpendicular to Its Plane:

  I = 2MR²

• Radius of Gyration:

  k = √2R

• About Its Diameter:

  Ix = Iy = MR²/2

• Radius of Gyration:

  k = R√2

• In Plane of the Ring, Parallel to Diameter:

  I = 3/2 MR²

• Radius of Gyration:

  k = 3R/√2

Angular Momentum

• Definition:
  • Angular momentum (L) = Moment of linear momentum
  • L = r × p
  • Magnitude: L = |r × p|
• Formula: L = mrv sinθ

Dimensions and Units:

• Dimensions: [Angular momentum] = [M1L2T⁻¹]
• SI Unit: Kilogram-metre²/second (kg·m²/s)

Expression for Angular Momentum

• In Terms of Moment of Inertia:
  • For a rigid object with N particles:
    • L = m₁r₁²ω + m₂r₂²ω + ... + mᴺrᴺ²ω
    • L = (m₁r₁² + m₂r₂² + ... + mᴺrᴺ²)ω
    • L = Iω
  • Where:
    • L = Angular momentum
    • I = Moment of inertia
    • ω = Angular velocity

Kinetic Energy of a Rotating Body

• Formula:
  • E = ½ Iω²
  • E = ½ (Iω)ω
  • E = ½ Lω

Physical Significance: The expression for angular momentum L = Iω is analogous to the expression p = mv of linear momentum, with moment of inertia I replacing mass.

Expression for Torque (in terms of Moment of Inertia)

• Torque (τ):
  • For a rigid body with N particles:
    • τ = τ₁ + τ₂ + ... + τᴺ
    • τ = (m₁r₁² + m₂r₂² + ... + mᴺrᴺ²)α
    • τ = Iα
  • Where: (τ = Torque, I = Moment of inertia, α = Angular acceleration)
   

Physical Significance:The relation τ = Iα is analogous to f = ma for translational motion, with moment of inertia I replacing mass.

Conservation of Angular Momentum

• Principle: Angular momentum of a body is conserved if the resultant external torque on the body is zero.
• Proof:
  • Angular momentum (l) is conserved if torque (τ) is zero.
  • If τ = 0, dl/dt = 0, meaning angular momentum is constant.

Examples of Conservation of Angular Momentum

1. Ballet Dancers:

   • Purpose: Increase rotation speed during spins.
   • Method: Reduce moment of inertia.
   • Actions:
     • Gather for smaller radius rounds.
     • Extend limbs for larger radius rounds.
   • Effect: Increases frequency for thrilling performance.

2. Divers:

   • Purpose: Increase rotation speed during somersaults.
   • Method: Manipulate moment of inertia.
   • Actions:
     • Stretch body on diving board.
     • Fold body in mid-air.
     • Streamline body for smooth water entry.
   • Effect: Increases rotation frequency for an attractive performance.

3. Physical Significance:

   • Product Relation: L = Iω = I(2πn) remains constant.
   • Increase Moment of Inertia: Decreases angular speed and frequency.
   • Decrease Moment of Inertia: Increases frequency.

Rolling Motion

• Two Motions of Pure Rolling:

  Circular motion and linear motion occur simultaneously.

• Description

  • Circularly symmetric rigid body (e.g., wheel or disc) rolls on a plane surface with friction.
  • Center of mass at geometric center O.

• Purely Translational Motion

  • Every point on the wheel has the same linear velocity as center O (v_{CM} = v_O).

• Purely Rotational Motion

  • Every point on the wheel rotates about the axis with angular velocity.
  • Each point on the rim has the same linear speed ωR.

• Combined Motion

  • Observed in inertial frame where surface is at rest.
  • No slipping; point of contact with surface is stationary (v_A = 0).
  • Wheel turns about instantaneous axis through point of contact A.

• Instantaneous Linear Speed

  • Point C (at the top) has linear speed v_C = ω(2R) faster than any other point.
    

Expression for Kinetic Energy of a Rolling Body:

• Rolling motion of body is translation of centre of mass (CM) and rotation about an axis through CM.
• Kinetic energy of rolling body = E = Etran + Erot (equation 1).
• Etran: Kinetic energy of translation of CM, Erot: Kinetic energy of rotation about axis through CM.
• M: mass, R: radius, ω: angular speed, k: radius of gyration, I: moment of inertia, v: translational speed.
• v = ωR, I = Mk² (equation 2).
• Etran = 1/2Mv², Erot = 1/2Iω² (equation 3).
• E = 1/2Mv² + 1/2Iω² = 1/2Mv² + 1/2Iv²R².
• E = 1/2Mv²(1 + IMR²) = 1/2Mv²(1 + Mk²MR²) = 1/2Mv²(1 + k²R²) (equations 4, 5).
• E = 1/2Mω²R²(1 + k²R²) = 1/2Mω²(R² + k²) (equation 6).
• Equation 4, 5, or 6 gives the required expression.

Kinetic Energy of a Solid Sphere:

• For solid sphere, k = √(2/5)R, I = (2/5)MR² (from k and I definitions).
• IMR² = 2/5 (from equation 7).
• Substituting in equation 4: E = (7/10)Mv² (equation 8).

Summary:

• Kinetic energy of a rolling body = E = Etran + Erot.
• Expressions derived using mass (M), radius (R), angular speed (ω), radius of gyration (k), moment of inertia (I), translational speed (v).
• Final expression for kinetic energy depends on either equation 4, 5, or 6.
• For solid sphere: k = √(2/5)R, I = (2/5)MR², giving E = (7/10)Mv².
  

Linear Acceleration and Speed While Pure Rolling Down an Inclined Plane:

• Circularly symmetric rigid body (e.g., sphere, wheel, disc) rolls with friction down inclined plane at angle θ to horizontal.
• M: mass, R: radius, I: moment of inertia, h: height, v: translational speed.
• Kinetic energy at bottom: E = 1/2Mv²(1+I/MR²) (equation 1).
• If k is radius of gyration, I = Mk².
• From conservation of energy: (KE + PE)initial = (KE + PE)final (equation 2).
• Mgh = 1/2Mv²(1+k²/R²) (equation 3).
• v² = 2gh(1+k²/R²).
• v = √(2gh(1+k²/R²)) (equation 4).
• Since h = s sin θ: v = √(2gssinθ(1+k²/R²)) (equation 5).
• Acceleration (a) along incline: a = v²/(2s) (equation 6).
• a = 2gssinθ/(1+k²/R²).
• a = gsinθ/(1+k²/R²) (equation 7).
• Time (t) taken to travel distance s: t = √(2s/a) = √(2s/gsinθ.(1+k²/R²)).

Summary:

• Friction allows rolling without slipping down inclined plane.
• Kinetic energy at bottom: E = 1/2Mv²(1+IMR²).
• Using conservation of energy: Mgh = 1/2Mv²(1+k²R²).
• Linear speed (v) and acceleration (a) derived using equations.
• Time (t) calculated for distance s.

Expression for the Speed of Various Bodies Rolling Down an Inclined Plane:

(i) Ring:

• For a ring: I = MR².
• k² = I/M = R².
• v = 2gh√(1+1) = gh√2.

(ii) Solid Cylinder or Disc:

• For solid cylinder: I = 1/2MR².
• k² = I/M = 1/2R².
• v = 2gh√(1+1/2) = (4/3)gh.

(iii) Spherical Shell (Hollow Sphere):

• For spherical shell: I = 2/3MR².
• k² = I/M = 2/3R².
• v = 2gh√(1+2/3) = (5/3)gh.

(iv) Solid Sphere:

• For solid sphere: I = 2/5MR².
• k² = I/M = 2/5R².
• v = 2gh√(1+2/5) = (10/7)gh.

Summary:

• Ring: v = gh√2.
• Solid Cylinder: v = (4/3)gh.
• Spherical Shell: v = (5/3)gh.
• Solid Sphere: v = (10/7)gh.
  

Expression for the Acceleration of Various Bodies Rolling Down an Inclined Plane:

(i) Ring:

• For a ring: I = MR².
• a = gsinθ/(1+1) = 0.5 g sin θ.

(ii) Solid Cylinder or Disc:

• For solid cylinder: I = 1/2MR².
• a = gsinθ/(1+1/2) = (2/3) g sin θ.

(iii) Spherical Shell (Hollow Sphere):

• For spherical shell: I = 2/3MR².
• a = gsinθ/(1+2/3) = (3/5) g sin θ.

(iv) Solid Sphere:

• For solid sphere: I = 2/5MR².
• a = gsinθ/(1+2/5) = (5/7) g sin θ.

Note:

• In descending order of acceleration: solid sphere > spherical shell > solid cylinder > ring.
• This means if released simultaneously from rest, solid sphere reaches bottom first, followed by spherical shell, solid cylinder, and lastly, the ring.