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# Rotational Motion Formulas list

This Rotational motion formulas list has a list of frequently used rotational motion equations. These equations involve trigonometry and vector products.
Rotational motion is the motion of a body around a fixed axis (see types of motion). Variables of motion in case of rotational motion are
1. angular displacement $\theta$
2. angular velocity $\omega$
3. angular acceleration $\alpha$
Also see translational motion

### Rotational motion equations formula list

If a body is executing rotation with constant acceleration, the equations of motion can be written as $\omega =\omega _0+\alpha t$ $\theta =\omega _0t+\frac{1}{2}\alpha t^2$ $\omega ^2-\omega _{0}^{2}=2\alpha t$ Units and notations used

• $\theta$ : angular displacement its unit is $radian$
• $\omega_0$: initial angular velocity its unit is $rad \,\, s^{-1}$
• $\omega$ : final angular velocity its unit is $rad \,\, s^{-1}$
• $\alpha$ : angular acceleration its unit is $rad \,\, s^{-2}$

### Formulas for torque, angular momentum, power and work done

Torque \begin{align*}\text{Torque} =&\text{force }\times \\ &\text{its perpendicular distance from axis of rotation}\end{align*} or, $\tau =Fd$ Torque $\tau =rF\sin \theta$ or, $\vec{\tau} =\vec{r}\times \vec{F}$

Power of Torque $\text{Power of a torque} = \text{torque} \times \text{angular velocity}$ or, $P=\tau \omega$ $\text{Work done by torque = torque} \times \text{ angular displacement}$ or, $W=\tau \theta$

Angular Momentum \begin{align*}\text{Angular momentum} =&\text{ Linear momentum } \times \\ &\text{its perpendicular distance from the axis of rotation}\end{align*} $L=pd$ $\text{Angular momentum } l=rp\sin \theta$ or, $\vec{L}=\vec{r}\times \vec{p}$ For a particle of mass $m$ moving with uniform speed $v$ along a circle of radius $r$, $L=mvr$ $\text{torque = rate of change of angular momentum}$ or, $\tau =\frac{dL}{dt}$

Unit used:
torque – $Nm$
Work done – $Joule$
Power – $Watt$
angular velocity – $\text{rad}.\text{s^{-1}}$
angular momentum – $Kgm^2s^{-1}$

### Moment of Inertia formula list

Moment of inertia of a body about any given axis of rotation, $I=m_1r_1^2+m_2r_2^2+m_3r_3^2+……..m_nr_n^2=\sum_{i=1}^n{m_ir_i^2}$ Radius of gyration $K$ is given by $K=\sqrt{\frac{I}{M}}$ Theorem of perpendicular axis $I_z=I_x+I_y$ Theorem of parallel axis $I=I_{CM}+Md^2$ Rotational Kinetic Energy $K.E.=\frac{1}{2}I\omega ^2$ Total Kinetic Energy = Rotational K.E. + Translational K.E. $\text{Total K.E.} = \frac{1}{2}I\omega ^2+\frac{1}{2}Mv^2$

Units and notations Used

### Relations between torque, angular momentum and M.I. $I$

Torque = M.I $\times$ angular acceleration
or,$\tau=I\alpha$ Work done by torque,$W=\tau \theta$ Angular momentum = M.I. $\times$ angular Velocity
or, $L=I\omega$

Unit and terms Used

• Torque $\tau$ is in $N\,\,m$.
• Moment of inertia $I$ is in $Kg\,\,m^2$.
• Angular momentum $L$ is in $Kg\,\,m^2s^{-1}$.

### Rolling without slipping

For a cylinder of mass $M$ and radius $R$, rolling motion without slipping down a plane inclined at an angle $\theta$ with the horizontal,
1. Force of friction between the plane and the cylinder
$f=\frac{1}{3}Mg\sin \theta$
2. Linear acceleration,
$a=\frac{2}{3}g \sin \theta$
3. Conditions for rolling without slipping is
$\mu_s > \frac{1}{3} \tan \theta$
Here $a$ and $g$ are in $m\,\,s^{-1}$ and $\mu_s$ has no units.

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