Rolling without Slipping on Incline:
$$a_{\text{cm}} = \frac{g \sin\theta}{1 + \frac{I_{\text{cm}}}{M R^2}} = \frac{g \sin\theta}{1 + k^2/R^2}$$
Values of $k^2/R^2$: Solid Cylinder $= 1/2$, Solid Sphere $= 2/5$, Hollow Sphere $= 2/3$, Ring/Hollow Cylinder $= 1$.
Escape Velocity & Orbital Speed:
$$v_e = \sqrt{\frac{2GM}{R}} = R\sqrt{\frac{8\pi G \rho}{3}}, \quad v_o = \sqrt{\frac{GM}{R}} = \frac{v_e}{\sqrt{2}}$$
Terminal Velocity (Stokes' Law):
$$v_t = \frac{2r^2 (\rho - \sigma) g}{9\eta} \implies v_t \propto r^2$$
Dielectric Capacitor:
$$C = \frac{\varepsilon_0 A}{d - t + \frac{t}{K}}$$
Rotating Rod Motional EMF:
$$\mathcal{E} = \frac{1}{2} B \omega L^2$$
Series LCR Quality Factor & Resonance:
$$\omega_0 = \frac{1}{\sqrt{LC}}, \quad Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 C R} = \frac{1}{R}\sqrt{\frac{L}{C}}$$
Lens Maker's Formula:
$$\frac{1}{f} = \left(\frac{\mu_g}{\mu_m} - 1\right) \left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$
YDSE Fringe Width:
$$\beta = \frac{\lambda D}{d}, \quad \text{Fringe Shift} = \frac{(\mu - 1)t D}{d}$$
de Broglie Wavelength of Charged Particle:
$$\lambda = \frac{h}{\sqrt{2 m q V}}$$
Gibbs-Helmholtz Spontaneity:
$$\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ = -RT \ln K_{eq}$$
First Order Kinetics Relations:
$$k = \frac{2.303}{t} \log\left(\frac{[A]_0}{[A]}\right), \quad t_{50\%} = \frac{0.693}{k}, \quad t_{99.9\%} = 10 \times t_{50\%}$$
Nernst Equation at 298 K:
$$E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{0.0591}{n} \log_{10} Q$$
Colligative Freezing Point & Osmotic Pressure:
$$\Delta T_f = i K_f m, \quad \pi = i C R T, \quad i = 1 + (n - 1)\alpha$$
Spin-Only Magnetic Moment:
$$\mu = \sqrt{n(n+2)}\text{ BM}$$
For $n = 1 \implies 1.73\text{ BM}$, $n = 2 \implies 2.83\text{ BM}$, $n = 3 \implies 3.87\text{ BM}$, $n = 4 \implies 4.90\text{ BM}$, $n = 5 \implies 5.92\text{ BM}$.
Molecular Orbital Theory Bond Order:
$$\text{Bond Order} = \frac{N_b - N_a}{2}$$
King's Definite Integral Property:
$$\int_a^b f(x)dx = \int_a^b f(a + b - x)dx$$
Leibniz Rule for Differentiation under Integral Sign:
$$\frac{d}{dx} \int_{u(x)}^{v(x)} f(t) dt = f(v(x)) v'(x) - f(u(x)) u'(x)$$
Linear Differential Equation Integrating Factor:
$$\frac{dy}{dx} + P(x) y = Q(x) \implies \text{I.F.} = e^{\int P(x)dx}, \quad y \cdot \text{I.F.} = \int Q(x) \cdot \text{I.F.} dx + C$$
Parabola Tangents & Director Line:
$$y = mx + \frac{a}{m} \quad \text{(Tangent to } y^2 = 4ax\text{)}, \quad \text{Director line is the Directrix: } x + a = 0$$
Ellipse & Hyperbola Director Circles:
$$\text{Ellipse: } x^2 + y^2 = a^2 + b^2, \quad \text{Hyperbola: } x^2 + y^2 = a^2 - b^2$$
Newton's Sum Theorem for Quadratic Roots:
$$a S_n + b S_{n-1} + c S_{n-2} = 0 \quad \text{where } S_n = \alpha^n \pm \beta^n$$
Adjoint Determinant Properties ($n \times n$ matrix):
$$\det(\operatorname{adj} A) = |A|^{n-1}, \quad \det(\operatorname{adj}(\operatorname{adj} A)) = |A|^{(n-1)^2}, \quad \det(kA) = k^n |A|$$
Scalar Triple Product (Coplanarity Condition):
$$[\vec{a}\ \vec{b}\ \vec{c}] = 0 \iff \vec{a}, \vec{b}, \vec{c} \text{ are coplanar}$$