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) :

• 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 :

• 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 :

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

Torque:

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

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

$ \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:

blic)

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

$|\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

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

(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 $A=\frac{F_{0}}{\omega_{d} b}$