Chapter 5: Circular Motion

• Angular Displacement ($\Delta\theta$)

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  • Angle subtended at the center by the arc length.
  • $\Delta\theta = \frac{\Delta s}{r}$.
  • Unit: Radian (rad).
  • Dimensionless.
  • Vector quantity (direction by right-hand rule).
  • Small angular displacements are vectors, large ones are scalars.
  • $1~radian = 57.3^\circ$.
  • $2\pi~rad = 360^\circ = 1~revolution$.

• Angular Velocity ($\omega$)

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  • Rate of change of angular displacement.
  • $\omega_{avg} = \frac{\Delta\theta}{\Delta t}$.
  • $\omega_{inst} = \frac{d\theta}{dt}$.
  • Unit: rad/s.
  • Dimensions: $[T^{-1}]$.
  • Vector quantity (direction by right-hand rule).

  • Relation between Linear and Angular Velocity

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    • $v = r\omega$.
    • In vector form: $\vec{v} = \vec{\omega} \times \vec{r}$.

  • Period (T) and Frequency (f)

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    • Period (T): Time for one complete revolution. Unit: second (s).
    • Frequency (f): Number of revolutions per second. Unit: Hertz (Hz) or $s^{-1}$.
    • $T = \frac{1}{f}$.
    • Relation with angular velocity: $\omega = \frac{2\pi}{T} = 2\pi f$.

• Angular Acceleration ($\alpha$)

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  • Rate of change of angular velocity.
  • $\alpha_{avg} = \frac{\Delta\omega}{\Delta t}$.
  • $\alpha_{inst} = \frac{d\omega}{dt}$.
  • Unit: rad/s$^2$.
  • Dimensions: $[T^{-2}]$.
  • Vector quantity.

  • Relation between Linear and Angular Acceleration

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    • $a_t = r\alpha$ (tangential acceleration).
    • $\vec{a}_t = \vec{\alpha} \times \vec{r}$.

• Centripetal Force ($F_c$)

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  • Force that keeps a body in circular path.
  • Always directed towards the center.
  • $F_c = \frac{mv^2}{r} = mr\omega^2 = m\omega v$.
  • Unit: Newton (N).
  • Work done by centripetal force is zero (force is perpendicular to displacement).

• Centripetal Acceleration ($a_c$)

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  • Acceleration directed towards the center of the circular path.
  • $a_c = \frac{v^2}{r} = r\omega^2 = v\omega$.
  • Unit: m/s$^2$.
  • Dimensions: $[LT^{-2}]$.

• Centrifugal Force

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  • Fictitious force, equal in magnitude to centripetal force but directed outwards.
  • Experienced in a rotating non-inertial frame of reference.

• Moment of Inertia (I)

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  • Rotational analogue of mass.
  • Measures resistance to angular acceleration.
  • For a point mass: $I = mr^2$.
  • For a system of particles: $I = \sum m_i r_i^2$.
  • Unit: $kg~m^2$.
  • Scalar quantity.

  • Moment of Inertia for Various Shapes

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    Shape	Axis	Moment of Inertia (I)
    Solid cylinder/disc	Through center, perpendicular to plane	$\frac{1}{2}MR^2$
    Thin hoop/ring	Through center, perpendicular to plane	$MR^2$
    Solid sphere	Through center	$\frac{2}{5}MR^2$
    Thin rod	Through center, perpendicular to length	$\frac{1}{12}ML^2$
    Thin rod	Through end, perpendicular to length	$\frac{1}{3}ML^2$

• Torque and Angular Acceleration

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  • Relationship between net torque and angular acceleration.
  • $\vec{\tau}_{net} = I\vec{\alpha}$.
  • Rotational analogue of Newton's second law ($F=ma$).

• Angular Momentum (L)

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  • Rotational analogue of linear momentum.
  • For a point mass: $\vec{L} = \vec{r} \times \vec{p} = \vec{r} \times (m\vec{v})$.
  • For a rigid body: $L = I\omega$.
  • Unit: J s or N m s.
  • Dimensions: $[ML^2T^{-1}]$.
  • Vector quantity.

  • Law of Conservation of Angular Momentum

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    • In an isolated system, total angular momentum remains constant.
    • If external torque is zero, $I_1\omega_1 = I_2\omega_2$.
    • Examples: Ice skater spinning, diver tucking in.

• Rotational Kinetic Energy (KE$_r$)

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  • Kinetic energy due to rotation.
  • $KE_r = \frac{1}{2}I\omega^2$.
  • Unit: Joule (J).
  • Dimensions: $[ML^2T^{-2}]$.
  • Relation with angular momentum: $KE_r = \frac{L^2}{2I}$.

• Artificial Gravity

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  • Simulated gravity in space stations via rotation.
  • Centripetal force provides the sensation of weight.
  • $g_{eff} = r\omega^2$.

• Banking of Roads

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  • Tilting of road curves to provide necessary centripetal force.
  • Reduces reliance on friction, safer at high speeds.

  • Banking Angle ($\theta$)

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    • $\tan\theta = \frac{v^2}{rg}$.
    • $v$: speed of vehicle.
    • $r$: radius of curve.
    • $g$: acceleration due to gravity.
    • Ideal banking angle: no friction required.

• Vertical Circular Motion

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  • Motion in a vertical circle (e.g., roller coaster loop).
  • Speed and tension vary throughout the motion due to gravity.

  • Minimum Speed at Top

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    • For object to complete loop: $v_{top} = \sqrt{gr}$.
    • Tension at top: $T_{top} = 0$ at minimum speed.

  • Speed at Bottom

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    • $v_{bottom} = \sqrt{5gr}$.
    • Tension at bottom: $T_{bottom} = 6mg$.

  • Speed at Mid-point (Horizontal)

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    • $v_{mid} = \sqrt{3gr}$.
    • Tension at mid-point: $T_{mid} = 3mg$.

• Orbital Velocity ($v_o$)

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  • Velocity required for an object to maintain a stable orbit around a celestial body.
  • For an object orbiting Earth at radius $r$: $v_o = \sqrt{\frac{GM}{r}}$.
  • For an object orbiting near Earth's surface ($r \approx R_e$): $v_o = \sqrt{gR_e}$.
  • Value near Earth's surface: $v_o \approx 7.9~km/s$.
  • Independent of the mass of the orbiting object.

• Quantity Translation Relation Table

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  Linear Quantity	Rotational Quantity
  Displacement ($s$)	Angular Displacement ($\theta$)
  Velocity ($v$)	Angular Velocity ($\omega$)
  Acceleration ($a$)	Angular Acceleration ($\alpha$)
  Mass ($m$)	Moment of Inertia ($I$)
  Force ($F$)	Torque ($\tau$)
  Linear Momentum ($p$)	Angular Momentum ($L$)
  Kinetic Energy ($\frac{1}{2}mv^2$)	Rotational Kinetic Energy ($\frac{1}{2}I\omega^2$)

• Analogy between Translatory and Rotatory Motion

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  Translatory Motion	Rotatory Motion
  Displacement ($\vec{s}$)	Angular Displacement ($\vec{\theta}$)
  Velocity ($\vec{v}$)	Angular Velocity ($\vec{\omega}$)
  Acceleration ($\vec{a}$)	Angular Acceleration ($\vec{\alpha}$)
  Mass ($m$)	Moment of Inertia ($I$)
  Force ($\vec{F}$)	Torque ($\vec{\tau}$)
  Linear Momentum ($\vec{p} = m\vec{v}$)	Angular Momentum ($\vec{L} = I\vec{\omega}$)
  Newton's 2nd Law: $\vec{F} = m\vec{a}$	Newton's 2nd Law: $\vec{\tau} = I\vec{\alpha}$
  Kinetic Energy: $KE = \frac{1}{2}mv^2$	Rotational Kinetic Energy: $KE_r = \frac{1}{2}I\omega^2$
  Work Done: $W = \vec{F} \cdot \vec{s}$	Work Done: $W = \vec{\tau} \cdot \vec{\theta}$
  Power: $P = \vec{F} \cdot \vec{v}$	Power: $P = \vec{\tau} \cdot \vec{\omega}$

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