Created By : Naaz Fatima

Reviewed By : Rajashekhar Valipishetty

Last Updated : May 10, 2023


Have you ever wondered how to understand the Rotational Motion concept in an easy way? You can look at the Rotational Motion Formulas provided here for quick reference. Master the concept of Rotational Motion by accessing the Rotational Motion Cheat Sheet & Tables here. Learn the formulas and implement them during your calculations and arrive at the solutions easily. Make use of the Physics Formulas existing to clear all your ambiguities.

Rotational Motion Formulae List

1. Angular displacement

╬╕ = \(\frac{arc}{radius}=\frac{\mathrm{s}}{\mathrm{r}}\) radian

2. Angular velocity

Average angular velocity
\(\bar{\omega}=\frac{\theta_{2}-\theta_{1}}{t_{2}-t_{1}}=\frac{\Delta \theta}{\Delta t}\) rad/s

Instantaneous angular velocity
ω = \(\frac{\mathrm{d} \theta}{\mathrm{dt}}\) rad/s
ω = 2πn = \(\left(\frac{2 \pi}{T}\right)\)

3. Angular acceleration

Average angular acceleration
\(\bar{\alpha}=\frac{\omega_{2}-\omega_{1}}{t_{2}-t_{1}}=\frac{\Delta \omega}{\Delta t}\) rad/s2

Instant angular acceleration
\(\alpha=\frac{d \omega}{d t}=\frac{d^{2} \theta}{d t^{2}}\)rad/s2

4. Relation between linear velocity and angular velocity

v = ωr = 2πnr = \(\frac{2 \pi r}{T}\), \(\overrightarrow{\mathbf{v}}=\vec{\omega} \times \vec{r}\)

5. Relation between linear acceleration and angular acceleration

a = \(\frac{d v}{d t}=r \frac{d \omega}{d t}\) = r╬▒
Tangential acceleration
\(\overrightarrow{a_{t}}=\vec{\alpha} \times \vec{r}\)

Radial acceleration
\(\overrightarrow{\mathrm{a}}_{\mathrm{r}}=\vec{\omega} \times \overrightarrow{\mathrm{v}}\)

Resultant acceleration
\(\vec{a}=\vec{a}_{t}+\vec{a}_{r}\)

6. Equations of rotational motion

ω2 = ω1 + αt
θ = ω1t + \(\frac{1}{2}\)αt2
ω22 = ω12 + 2αθ
╬╕nth = ╧ë1 + \(\frac{\alpha}{2}\) (2n – 1)

7. Moment of inertia

I = m1r12 + m2r22 + m3r32 + = \(\sum_{i=1}^{n} m_{i}{\mathbf{r}}_{i}^{2}\)
For a body with uniform mass distribution
I = Γê½ r2 dm

8. Theorems of moment of inertia

  • Theorem of perpendicular axes Iz = Ix + Iy. It is applicable only for lamina(planar body).
  • Theorem of parallel axes –
    I = IG + Md2 = MK2 + Md2

9. Moment of inertia of diatomic molecule

I = m1r12 + m2r22 = \(\left(\frac{m_{1} m_{2}}{m_{1}+m_{2}}\right)\) r2 = ┬╡r2
Rotational Motion formulas img 1

10. Moment of inertia of some objects

Ring:
I = MR2(axis)
I = MR2/2 (diameter)
I = 2 MR2 (tangential to rim, parallel to axis)
I = (3 /2)MR2 (tangential to rim, parallel to diameter)

Disc:
I = \(\frac{1}{2}\) MR2 (axis)
Rotational Motion formulas img 2
I = \(\frac{1}{4}\) MR2 (diameter)
Rotational Motion formulas img 3
I = \(\frac{3}{2}\) MR2 (tangential to rim, parallel to axis)
Rotational Motion formulas img 4
I = \(\frac{5}{4}\) MR2 (tangential to rim, parallel to diameter)
Rotational Motion formulas img 5

Cylinder:
About axis I = \(\frac{1}{2}\)MR2
Rotational Motion formulas img 6
perpendicular the length and passing through C.M.
I = \(\frac{M L^{2}}{12}+\frac{M R^{2}}{4}\)
Rotational Motion formulas img 7

Thin rod:
I = \(\frac{1}{12}\)ML2 (about centre)
Rotational Motion formulas img 8
I = \(\frac{1}{3}\)ML2 (about one end)
Rotational Motion formulas img 9

Hollow sphere:
Idia = \(\frac{2}{3}\)MR2
Rotational Motion formulas img 10
Itangential = \(\frac{5}{3}\)MR2
Rotational Motion formulas img 11

Solid sphere:
Idia = \(\frac{2}{5}\)MR2
Rotational Motion formulas img 12
Itangential = \(\frac{7}{5}\)MR2
Rotational Motion formulas img 13

Rectangular plate
Ic = \(\frac{M\left(L^{2}+B^{2}\right)}{12}\)
Rotational Motion formulas img 14

Cube:
I = \(\frac{1}{6}\)Ma2
Rotational Motion formulas img 15

Annular disc
I = \(\frac{1}{2}\)M(R12 + R22)
Rotational Motion formulas img 16

Right circular cone:
I = \(\frac{3}{10}\)MR2
Rotational Motion formulas img 17

Triangular Lamina:
I = \(\frac{1}{6}\)MH2(about base axis)
Rotational Motion formulas img 18

11. Radius of gyration

I = MK2 = \(\sum_{i=1}^{n} m_{i} r_{i}^{2}\)
K = \(\left(\frac{\mathrm{I}}{\mathrm{M}}\right)^{1 / 2}=\left[\frac{\sum \mathrm{m}_{\mathrm{i}} \mathrm{r}_{\mathrm{i}}^{2}}{\mathrm{M}}\right]^{1 / 2}\)
Rotational Motion formulas img 19

12. Torque

\(\vec{\tau}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{F}}\); \(|\vec{\tau}|\) = rFsinθ, τ = Iα
Work done by torque W = \(\int_{\theta_{1}}^{\theta_{2}} \tau \mathrm{d} \theta\)
Work done in twisting a wire W = \(\int_{0}^{\theta}(\mathrm{c} \theta) \mathrm{d} \theta=\frac{1}{2} \mathrm{c} \theta^{2}\)

13. Angular momentum

\(\overrightarrow{\mathrm{L}}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{p}}=m(\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{v}})\)
\(|\overrightarrow{\mathrm{L}}|\) = rp sin ╬╕
\(\overrightarrow{\mathrm{L}}=\mathrm{I} \vec{\omega}\) and \(\vec{\tau}=\frac{d \vec{L}}{d t}\)
In absence of external torque i.e., \(\vec{\tau}\) = 0,
so that L = Iω = constant, which is law of conservation of angular momentum.

14. Angular impulse

ΔL = \(\int_{0}^{t} \tau \mathrm{dt}\)

15. Rotational kinetic energy

Erot = \(\frac{1}{2}\) Iω2 = \(\frac{\mathrm{L}^{2}}{2 \mathrm{I}}\)
or \(\frac{\mathrm{L}^{2}}{2 \mathrm{I}}=\frac{\overrightarrow{\mathrm{L}} \cdot \overrightarrow{\mathrm{L}}}{2 \mathrm{I}}\)

Rotational power

Prot = \(\vec{\tau} \cdot \vec{\omega}\)
Etotal = Etrans + Erot = \(\frac{1}{2}\)Mv2 + \(\frac{1}{2}\) Iω2
Etotal = \(\frac{1}{2}\)Mv2\(\left(1+\frac{\mathrm{K}^{2}}{\mathrm{R}^{2}}\right)\)

16. Motion on an inclined plane
(A) For translational motion only (without rotation)
\(\frac{1}{2}\) mv2 = mgh, velocity v = \(\sqrt{2 \mathrm{gh}}=\sqrt{2 \mathrm{gs} \sin \theta}\)
Acceleration f = g sin ╬╕, time t = \(\left(\frac{2 s}{g \sin \theta}\right)^{1 / 2}\)

(B) For rolling motion (translational + rotational)
Rotational Motion formulas img 20
\(\frac{1}{2}\)mv2 \(\left(1+\frac{K^{2}}{R^{2}}\right)\) = mgh
Velocity v \(=\left[\frac{2 g \sin \theta}{\left(1+\frac{K^{2}}{R^{2}}\right)}\right]^{1 / 2}=\left[\frac{2 g h}{\left(1+\frac{K^{2}}{R^{2}}\right)}\right]^{1 / 2}\)
Acceleration f = \(\frac{g \sin \theta}{\left(1+\frac{K^{2}}{R^{2}}\right)}\)
Time t = \(\sqrt{\frac{2 \mathrm{s}}{\mathrm{f}}}\) = \(\left[\frac{2 s\left(1+\frac{K^{2}}{R^{2}}\right)^{1 / 2}}{g \sin \theta}\right]^{1 / 2}\)

17. Motion of a body tied to a string wrapped on a cylinder

Rotational Motion formulas img 21
Linear acceleration of the body a = \(\frac{g}{\left(1+\frac{I}{m r^{2}}\right)}\)
Tension in the string T = \(\frac{m g}{\left(1+\frac{m r^{2}}{I}\right)}\)
Angular acceleration of the cylinder
╬▒ = \(\frac{a}{r}=\frac{g}{r\left(1+\frac{I}{m r^{2}}\right)}\)

18. Compound pendulum
Rotational Motion formulas img 22
Periodic time
T = 2π\(\left[\frac{\frac{K^{2}}{\ell}+\ell}{g}\right]^{1 / 2}\) = 2π\(\sqrt{\frac{\mathrm{I}}{\mathrm{Mg} \ell}}\)
T = 2π \(\left[\frac{L}{g}\right]^{1 / 2}\)
I = M(K2 + l2)
K → Radius of gyration about C.M.
Length of equivalent simple pendulum
L = \(\left(\frac{K^{2}}{\ell}+\ell\right)\)
Tmin =2π\(\sqrt{\frac{2 \mathrm{K}}{\mathrm{g}}}\)

19. Centre of mass

About centre of mass \(\sum_{i=1}^{n} m_{i} \vec{r}_{i}\) = 0
Position of centre of mass
\(\overrightarrow{\mathrm{r}}_{\mathrm{cm}}=\frac{\mathrm{m}_{1} \overrightarrow{\mathrm{r}}_{1}+\mathrm{m}_{2} \overrightarrow{\mathrm{r}}_{2}+\ldots .+\mathrm{m}_{\mathrm{n}} \mathrm{r}_{\mathrm{n}}}{\mathrm{m}_{1}+\mathrm{m}_{2}+\ldots .+\mathrm{m}_{\mathrm{n}}}\)

Velocity of centre of mass
\(\overrightarrow{\mathrm{v}}_{\mathrm{cm}}=\frac{\sum_{i=1}^{n} m_{i} \vec{v}_{i}}{\sum_{i=1}^{n} m_{i}}=\frac{\overrightarrow{\mathrm{P}}}{\mathrm{M}}\)

20. C.M. for two particles system

\(\overrightarrow{\mathrm{X}}_{\mathrm{cm}}=\frac{\mathrm{m}_{1} \overrightarrow{\mathrm{x}}_{1}+\mathrm{m}_{2} \overrightarrow{\mathrm{x}}_{2}}{\mathrm{m}_{1}+\mathrm{m}_{2}}\),
\(\overrightarrow{\mathrm{y}}_{\mathrm{cm}}=\frac{\mathrm{m}_{1} \overrightarrow{\mathrm{y}}_{1}+\mathrm{m}_{2} \overrightarrow{\mathrm{y}}_{2}}{\mathrm{m}_{1}+\mathrm{m}_{2}}\),
\(\overrightarrow{\mathbf{z}}_{\mathrm{cm}}=\frac{\mathrm{m}_{1} \overrightarrow{\mathrm{z}}_{1}+\mathrm{m}_{2} \overrightarrow{\mathrm{z}}_{2}}{\mathrm{m}_{1}+\mathrm{m}_{2}}\)
\(\overrightarrow{\mathrm{V}}_{\mathrm{cm}}=\frac{\mathrm{m}_{1} \overrightarrow{\mathrm{v}}_{1}+\mathrm{m}_{2} \overrightarrow{\mathrm{v}}_{2}}{\mathrm{m}_{1}+\mathrm{m}_{2}}\),
\(\overrightarrow{\mathrm{a}}_{\mathrm{cm}}=\frac{\mathrm{m}_{1} \overrightarrow{\mathrm{a}}_{1}+\mathrm{m}_{2} \overrightarrow{\mathrm{a}}_{2}}{\mathrm{m}_{1}+\mathrm{m}_{2}}\)

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