Rigid Body :

If the above body is rigid

$$ V_A \cos \theta_1=V_B \cos \theta_2 $$

$V_{B A}=$ relative velocity of point $B$ with respect to point $A$.

Moment Of Inertia(I) :

PYQ-2023-Rotational-Motion-Q1, PYQ-2023-Rotational-Motion-Q2, PYQ-2023-Rotational-Motion-Q5, PYQ-2023-Rotational-Motion-Q6, PYQ-2023-Rotational-Motion-Q13, PYQ-2023-Rotational-Motion-Q15

• Definition : Moment of Inertia is defined as the capability of system to oppose the change produced in the rotational motion of a body.

• Moment of Inertia is a scalar positive quantity.

$$I =mr_{1}^{2}+m_{2}r_{2}^{2}+\ldots$$

$$I=I_1+I_2+I_3+\ldots$$

• SI units of Moment of Inertia is $\mathrm{Kgm}^{2}$.

Moment Of Inertia Of Different Object:

• A single particle : $$\mathrm{I}=\mathrm{mr}^{2}$$

  where: $m=$ mass of the particle

  $r=$ perpendicular distance of the particle from the axis about which moment of Inertia is to be calculated

• For many particles (system of particles) :

  $$I=\sum_{i=1}^{n} m_{i} r_{i}^{2}$$

• For a continuous object :

  $$\mathrm{I}=\int \mathrm{dmr} \mathrm{r}^{2}$$

  where $\mathrm{dm}=$ mass of a small element

  $r=$ perpendicular distance of the particle from the axis

• For a larger object :

  $$\mathrm{I}=\int \mathrm{dI}_{\text {element }}$$

  where: $\mathrm{dI}=$ moment of inertia of a small element

Two Important Theorems On Moment Of Inertia :

PYQ-2023-Rotational-Motion-Q1, PYQ-2023-Rotational-Motion-Q6, PYQ-2023-Rotational-Motion-Q15

• Perpendicular Axis Theorem

  [Only applicable to plane lamina (that means for 2-D objects only)].

  When object is in $x-y$ plane: $$I_{z}=I_{x}+I_{y}$$

• Parallel Axis Theorem

  (Applicable to any type of object):

  $$I_{AB}=I_{cm}+Md^{2}$$

  List of some useful formula :

Radius Of Gyration :

PYQ-2023-Rotational-Motion-Q1, PYQ-2023-Rotational-Motion-Q14

$$\mathrm{I}=\mathrm{MK}^{2}$$

Torque:

PYQ-2023-Rotational-Motion-Q4

$$\vec{\tau}=\vec{r} \times \vec{F}$$

Relation between ’ $\tau$ ’ and ’ $\alpha$ ’ (for hinged object or pure rotation)

$$ \vec{\tau} _{ext/Hinge } = I _{Hinge} \vec{\alpha}$$

Where: $\vec{\tau} _{ext/Hinge }$= net external torque acting on the body about Hinge point

$\mathrm{I} _{\text {Hinge }}=$ moment of Inertia of body about Hinge point

$$F_{1t}=M_{1} a_{1t}=M_1 r_1 \alpha$$

$$F_{2 t}=M_{2} a_{2 t}=M_{2} r_{2} \alpha$$

$$\tau_{resultant}=F_{1t} r_{1}+F_{2t} r_{2}+\ldots$$

$$=M_{1} \alpha r_{1}^{2}+M_{2} \alpha r_{2}^{2}+$$

$$\tau_{\text {resultant/ external}}=\mathrm{I} \alpha$$

Rotational Kinetic Energy

PYQ-2023-Rotational-Motion-Q11

$$ \text{K.E}=\frac{1}{2} I \omega^{2}$$

$$\vec{P}=M\vec{v} _{CM} \Rightarrow \vec{F} _{external} = M \vec{a} _{CM} $$

Net external force acting on the body has two parts tangential and centripetal.

$$ \Rightarrow F _C = ma _C = m \frac{v^2}{r _{CM}}=m \omega^{2} r _{cm} $$

$$ \Rightarrow F _t = ma _t = m\alpha r _{CM}$$

Rotational Equilibrium :

For translational equilibrium:

$$\Sigma F_{x}=0 \hspace{10mm}…(i)$$

$$\Sigma \mathrm{F}_{\mathrm{y}}=0 \hspace{10mm}…(ii)$$

The condition of rotational equilibrium is:

$$\Sigma \Gamma_{z}=0$$

Angular Momentum $(\vec{L})$:

• Angular Momentum Of A Particle About A Point:

$$\vec{L} =\vec{r} \times \vec{P} \quad \Rightarrow \quad L=rp \sin \theta$$

$$|\vec{L}| =r_{\perp} \times P $$

$$|\vec{L}| =P_{\perp} \times r$$

• Angular Momentum Of A Rigid Body Rotating About Fixed Axis :

$$\vec{L} _{H} = I _{H} \vec{\omega}$$

• $\mathrm{L}_{\mathrm{H}}=$ angular momentum of object about axis $\mathrm{H}$.

• $\mathrm{I}_{\mathrm{H}}=$ Moment of Inertia of rigid object about axis $\mathrm{H}$.

• $\omega=$ angular velocity of the object.

• Conservation of Angular Momentum:

  Angular momentum of a particle or a system remains constant if $\tau_{\mathrm{ext}}=0$ about that point or axis of rotation.

• Relation between Torque and Angular Momentum

  $$\vec{\tau}=\frac{\mathrm{d} \vec{\mathrm{L}}}{\mathrm{dt}}$$

  Torque is change in angular momentum.

• Impulse of Torque :

  $$\int \tau dt=\Delta J$$

  Where: $\Delta J$ is Change in angular momentum.

  For a rigid body, the distance between the particles remain unchanged during its motion i.e. $\mathrm{r}_{\mathrm{P} / \mathrm{Q}}=$ constant

• For velocities:

$$V_P=\sqrt{V_Q^{2}+(\omega r)^{2}+2 V_Q \omega r \cos \theta}$$

• For acceleration :

$\theta, \omega, \alpha$ are same about every point of the body (or any other point outside which is rigidly attached to the body).

Dynamics :

$$\vec{\tau} _{cm}=I _{cm} \vec{\alpha},$$

$$\vec F _{ext} = M \vec{a} _{cm}$$

$$\vec{P} _{system}=M \vec{v} _{cm}$$

$$\text{Total K.E.}=\frac{1}{2} M _{\mathrm{cm}^{2}}+\frac{1}{2} \mathrm{I} _{\mathrm{cm}} \omega^{2}$$

Angular momentum axis: $$A B=\vec{L} _{\text{about C.M.}} + \vec{L} _{\text {of C.M. about A B}}$$

$$\vec{L} _{AB}= I _{cm}\vec{\omega}+\vec{r _{cm}}\times M\vec{v} _{cm}$$

Simple Harmonic Motion

PYQ-2023-Oscillations-Q5

S.H.M.

$$\mathrm{F}=-\mathrm{kx}$$

General equation of S.H.M. is $x=A \sin (\omega t+\phi) ;(\omega t+\phi)$ is phase of the motion and $\phi$ is initial phase of the motion.

• Angular Frequency $(\omega)$ :

$$\omega=\frac{2 \pi}{T}=2 \pi f$$

• Time period $(\mathrm{T})$:

$$\mathrm{T}=\frac{2 \pi}{\omega}=2 \pi \sqrt{\frac{\mathrm{m}}{\mathrm{k}}}$$

• Speed :$$v=\omega \sqrt{A^{2}-x^{2}} $$

• Acceleration : $$ a=-\omega^{2} x$$

• Kinetic Energy (KE): $$\text{K.E}= \frac{1}{2} m v^{2}=\frac{1}{2} m \omega^{2}\left(A^{2}-x^{2}\right)=\frac{1}{2} k\left(A^{2}-x^{2}\right)$$

• Potential Energy (PE) : $$\text{ P.E}= \frac{1}{2} \mathrm{Kx}{ }^{2}$$

• Total Mechanical Energy (TME)

$$\text{T.E = K.E. + P.E.}=\frac{1}{2} k\left(A^{2}-x^{2}\right)+\frac{1}{2} K x^{2}=\frac{1}{2} K A^{2} = \text{constant}$$

Spring-Mass System

PYQ-2023-Oscillations-Q1, PYQ-2023-Oscillations-Q3

(1)

$$\Rightarrow T=2 \pi \sqrt{\frac{m}{k}}$$

(2)

$$T=2 \pi \sqrt{\frac{\mu}{K}}$$

where: $\mu=\frac{m_1 m_2}{\left(m_1+m_2\right)}$ is known as reduced mass

Combination Of Springs

• Series Combination : $$1 / k_{eq}=1 / k_{1}+1 / k_{2}$$

• Parallel combination : $$k_{eq}=k_1+k_2$$

Simple Pendulum:

$$T=2 \pi \sqrt{\frac{\ell}{g}}=2 \pi \sqrt{\frac{\ell}{g_{\text {eff. }}}}$$

In accelerating Reference Frame $g_{\text {eff }}$ is net acceleration due to pseudo force and gravitational force.

Compound Pendulum / Physical Pendulum:

$$T=2 \pi \sqrt{\frac{\mathrm{I}}{\mathrm{mg} \ell}}$$

where, $\mathrm{I}=\mathrm{I}_{\mathrm{CM}}+\mathrm{m} \ell^{2} ; \ell$ is distance between point of suspension and centre of mass.

Torsional Pendulum:

$$T=2 \pi \sqrt{\frac{I}{C}} \quad$$

where, $C=$ Torsional constant

Superposition of SHM’s along the same direction

$$x_{1}=A_{1} \sin \omega t$$

$$x_{2}=A_{2} \sin (\omega t+\theta)$$

If equation of resultant $\mathrm{SHM}$ is taken as $$\mathrm{x}=\mathrm{A} \sin (\omega \mathrm{t}+\phi)$$

$$A=\sqrt{A_{1}^{2}+A_{2}^{2}+2 A_{1} A_{2} \cos \theta}$$

$$\tan \phi=\frac{A_{2} \sin \theta}{A_{1}+A_{2} \cos \theta}$$

Damped Oscillation

• Damping force

$$\vec{\mathrm{F}}=-\mathrm{b} \vec{\mathrm{v}}$$

• Equation of motion

$$\frac{\mathrm{mdv}}{\mathrm{dt}}=-\mathrm{kx}-\mathrm{bv}$$

where:

• $b^{2}-4 m K>0$ over damping

• $b^{2}-4 m K=0$ critical damping

• $b^{2}-4 m K<0$ under damping

• For small damping the solution is of the form.

$$x=\left(A_{0} e^{-b t / 2 m}\right) \sin \left[\omega^{\prime} t+\delta\right]$$

where $\omega^{\prime}=\sqrt{\left(\frac{k}{m}\right)-\left(\frac{b}{2 m}\right)^{2}}$

For small b

• Angular Frequency: $$\omega^{\prime} \approx \sqrt{\mathrm{k} / \mathrm{m}},=\omega_{0}$$

• Amplitude: $$A=A_{0} e^{\frac{-b t}{2 m}}$$

• Energy $$E(t)=\frac{1}{2} K A^{2} e^{-b t / m}$$

• Quality factor or $Q$ value: $$Q=2 \pi \frac{E}{|\Delta E|}=\frac{\omega^{\prime}}{2 \omega_{Y}}$$

where $, \omega^{\prime}=\sqrt{\frac{k}{m} \cdot \frac{b^{2}}{4 m^{2}}} \quad, \omega_{Y}=\frac{b}{2 m}$

Forced Oscillations And Resonance

External Force $$F(t)=F_{0} \cos \omega_{d} t$$

$$x(t)=A \cos \left(\omega_{d} t+\phi\right)$$

$$A=\frac{F_{0}}{\sqrt{\left(m^{2}\left(\omega^{2}-\omega_{d}^{2}\right)^{2}+\omega_{d}^{2} b^{2}\right)}}$$

$$\tan \phi=\frac{-v_{0}}{\omega_{d} x_{0}}$$

(a) Small Damping $$A=\frac{F_{0}}{m\left(\omega^{2}-\omega_{d}^{2}\right)}$$

(b) Driving Frequency Close to Natural Frequency $$A=\frac{F_{0}}{\omega_{d} b}$$