Rotational Motion Formulas

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

╧ë₂ = ╧ë₁ + ╬▒t
╬╕ = ╧ë₁t + \(\frac{1}{2}\)╬▒t²
╧ë₂² = ╧ë₁² + 2╬▒╬╕
╬╕_n^th = ╧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 I_z = I_x + I_y. It is applicable only for lamina(planar body).
• Theorem of parallel axes –
  I = I_G + 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

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)

I = \(\frac{1}{4}\) MR² (diameter)

I = \(\frac{3}{2}\) MR² (tangential to rim, parallel to axis)

I = \(\frac{5}{4}\) MR² (tangential to rim, parallel to diameter)

Cylinder:
About axis I = \(\frac{1}{2}\)MR²

perpendicular the length and passing through C.M.
I = \(\frac{M L^{2}}{12}+\frac{M R^{2}}{4}\)

Thin rod:
I = \(\frac{1}{12}\)ML² (about centre)

I = \(\frac{1}{3}\)ML² (about one end)

Hollow sphere:
I_dia = \(\frac{2}{3}\)MR²

I_tangential = \(\frac{5}{3}\)MR²

Solid sphere:
I_dia = \(\frac{2}{5}\)MR²

I_tangential = \(\frac{7}{5}\)MR²

Rectangular plate
I_c = \(\frac{M\left(L^{2}+B^{2}\right)}{12}\)

Cube:
I = \(\frac{1}{6}\)Ma²

Annular disc
I = \(\frac{1}{2}\)M(R₁² + R₂²)

Right circular cone:
I = \(\frac{3}{10}\)MR²

Triangular Lamina:
I = \(\frac{1}{6}\)MH²(about base axis)

11. Radius of gyration

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

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

E_rot = \(\frac{1}{2}\) I╧ë² = \(\frac{\mathrm{L}^{2}}{2 \mathrm{I}}\)
or \(\frac{\mathrm{L}^{2}}{2 \mathrm{I}}=\frac{\overrightarrow{\mathrm{L}} \cdot \overrightarrow{\mathrm{L}}}{2 \mathrm{I}}\)

Rotational power

P_rot = \(\vec{\tau} \cdot \vec{\omega}\)
E_total = E_trans + E_rot = \(\frac{1}{2}\)Mv² + \(\frac{1}{2}\) I╧ë²
E_total = \(\frac{1}{2}\)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)

\(\frac{1}{2}\)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

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

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 = \(\left(\frac{K^{2}}{\ell}+\ell\right)\)
T_min =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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