Formula Quick Reference¶
A single-page cheat-sheet of every governing equation used across the 18 experiments, grouped the same way as the main Contents. Use this page while doing calculations so you don't have to open each experiment page individually — click the linked experiment name if you need the full derivation or context.
Rotational Dynamics¶
\[
T = 2\pi\sqrt{\dfrac{I l}{Mgb^{2}}} \qquad I = \dfrac{M g b^{2} T^{2}}{4\pi^{2} l}
\]
\[
C = \dfrac{\pi\eta r^{4}}{2l} \qquad I = \dfrac{2md^{2}T^{2}}{T'^{2}-T^{2}} \qquad \eta = \dfrac{8\pi l I}{r^{4}T^{2}}
\]
\[
\omega = \dfrac{2\pi n_{1}}{t} \qquad I = \dfrac{mr^{2}(2gh-\omega^{2}r^{2})}{\omega^{2}}\times\dfrac{n_{2}}{n_{1}+n_{2}}
\]
\[
T = 2\pi\sqrt{\dfrac{k^{2}+h^{2}}{gh}} \qquad h_{1}h_{2}=k^{2} \qquad g = \dfrac{4\pi^{2}L}{T^{2}}
\]
Elasticity of Solids¶
5. Non-Uniform (Cantilever) Bending
\[
Y = \dfrac{4Wl^{3}}{bd^{3}y}
\]
\[
Y = \dfrac{3Mga l^{2}}{2bd^{3}e}
\]
\[
Y = \dfrac{MgL}{\pi r^{2} l_{\text{ext}}}
\]
\[
\theta = \dfrac{2Cl}{\pi\eta r^{4}} \qquad \eta = \dfrac{2Cl}{\pi r^{4}\theta}
\]
Fluid Properties: Viscosity & Surface Tension¶
\[
\eta = \dfrac{\pi P r^{4} t}{8Vl}, \qquad P=\rho g h
\]
\[
\eta = \dfrac{2}{9}\dfrac{r^{2}g(\rho-\sigma)}{v}
\]
\[
T = \dfrac{h r \rho g}{2}
\]
Oscillations, Damping & Acoustic Methods (PG)¶
\[
f_{s}=\dfrac{1}{2\pi}\sqrt{\dfrac{g}{l}} \qquad f_{a}=\dfrac{1}{2\pi}\sqrt{\dfrac{g}{l}+\dfrac{2k}{m}} \qquad T_{\text{beat}}=\dfrac{1}{f_{a}-f_{s}}
\]
\[
A_{n}=A_{0}e^{-\gamma t} \qquad \delta = \ln\!\left(\dfrac{A_{n}}{A_{n+1}}\right)=\gamma T \qquad Q\approx\dfrac{\pi}{\delta}
\]
14. Forced Oscillations & Resonance
\[
A(\omega)=\dfrac{F_{0}/m}{\sqrt{(\omega_{0}^{2}-\omega^{2})^{2}+(2\gamma\omega)^{2}}} \qquad Q=\dfrac{f_{0}}{\Delta f}
\]
\[
\lambda = 2\overline{\Delta d} \qquad v=f\lambda \qquad \beta=\dfrac{1}{\rho v^{2}} \qquad K=\rho v^{2}
\]
Practical Applications¶
\[
v = \dfrac{M+m}{m}\sqrt{2gh} \qquad 1-\cos\theta = \dfrac{m^{2}v^{2}}{2gL(M+m)^{2}}
\]
17. Rolling Race on an Incline
\[
a = \dfrac{g\sin\theta}{1+k} \qquad I = kmr^{2} \qquad t = \sqrt{\dfrac{2L}{a}}
\]
\[
L = I\omega = \tfrac{1}{2}MR^{2}\omega \qquad \tau = Mgd \qquad \Omega = \dfrac{\tau}{L} = \dfrac{2gd}{R^{2}\omega}
\]