Class 12 Physics: Top 100 Most Important Numericals for Board Exams & MHT-CET

Physics Exam Strategy: Formulas & Solved Numericals

Success in the HSC Board and MHT-CET Physics papers relies heavily on mastering numericals. With 36 Marks dedicated solely to numerical problems, we have compiled the most important formulas and a model solved example for every single chapter.

How to use this guide:

For each chapter below, first memorize the "Key Formulas" list. Then, study the "Solved Example" to understand how to apply the values, convert units, and write the final answer with proper units.

1. Rotational Dynamics

Key Formulas

• Moment of Inertia: $I = \sum mr^2 = mk^2$.
• Parallel Axis: $I_o = I_c + Mh^2$.
• Perpendicular Axis: $I_z = I_x + I_y$.
• Banking Angle: $\tan\theta = \frac{v^2}{rg}$.
• Max Safe Speed: $v = \sqrt{\mu rg}$.

Solved Example Q. A racing car races around a circular track of radius 300 m. If coefficient of friction is 0.8, find max safe speed. ($g=9.8$)

Given: $r=300$, $\mu=0.8$

Formula: $v_{max} = \sqrt{\mu r g}$

Calc: $\sqrt{0.8 \times 300 \times 9.8} = \sqrt{2352}$

Ans: $v \approx 48.5$ m/s

2. Mechanical Properties of Fluids

Key Formulas

• Pressure: $P = h\rho g$.
• Surface Tension: $T = F/l$.
• Surface Energy: $W = T(dA)$.
• Excess Pressure (Bubble): $P_i - P_o = 4T/r$.
• Terminal Velocity: $v = \frac{2r^2(\rho-\sigma)g}{9\eta}$.

Solved Example Q. Calculate work done in blowing a soap bubble from radius 2 cm to 4 cm. ($T=0.03$ N/m)

Given: Soap bubble has 2 surfaces.

$W = T \times 2 \times (A_2 - A_1)$

$W = 0.03 \times 8\pi [(0.04)^2 - (0.02)^2]$

Ans: $9.05 \times 10^{-4}$ J

[Image of surface tension molecular forces]

3. KTG & Radiation

Key Formulas

• Ideal Gas Eq: $PV = nRT$.
• RMS Speed: $v_{rms} = \sqrt{3RT/M_0}$.
• Pressure: $P = \frac{1}{3}\rho v_{rms}^2$.
• Stefan's Law: $Q/t = \sigma A T^4$.

Solved Example Q. Calculate RMS speed of Oxygen at 27°C. ($M_0 = 32$g, $R=8.314$)

$T = 27+273 = 300$ K

$v_{rms} = \sqrt{\frac{3 \times 8.314 \times 300}{32 \times 10^{-3}}}$

$\sqrt{233831} \approx 483.5$

Ans: $483.56$ m/s

4. Thermodynamics

Key Formulas

• First Law: $Q = \Delta U + W$.
• Work (Isobaric): $W = P(V_2 - V_1)$.
• Adiabatic Work: $W = \frac{nR(T_1-T_2)}{\gamma-1}$.
• Efficiency: $\eta = 1 - T_C/T_H$.

Solved Example Q. Carnot engine operates between 327°C and 27°C. Find efficiency.

$T_H = 600$ K, $T_C = 300$ K

$\eta = 1 - (300/600) = 1 - 0.5$

Ans: 50% Efficiency

5. Oscillations

Key Formulas

• Diff Eq: $\frac{d^2x}{dt^2} + \omega^2 x = 0$.
• Velocity: $v = \omega \sqrt{A^2 - x^2}$.
• Period: $T = 2\pi / \omega$.
• Pendulum: $T = 2\pi \sqrt{L/g}$.

Solved Example Q. Period=2s, Amp=10cm. Find velocity at x=6cm.

$\omega = 2\pi/T = \pi$

$v = \pi \sqrt{10^2 - 6^2} = \pi \sqrt{64}$

$v = 3.142 \times 8$

Ans: $25.136$ cm/s

6. Superposition of Waves

Key Formulas

• Wave Eq: $y = A\sin(kx - \omega t)$.
• String Freq: $n = \frac{1}{2L}\sqrt{T/m}$.
• Closed Pipe: $n, 3n, 5n...$
• Beats: $N = |n_1 - n_2|$.

Solved Example Q. Frequencies 320 Hz and 324 Hz are sounded. Find beat period.

Beat Freq $N = 324 - 320 = 4$ Hz

Period $T = 1/N = 1/4$

Ans: 0.25 seconds

7. Wave Optics

Key Formulas

• Snell's Law: $\mu_1 \sin i = \mu_2 \sin r$.
• Fringe Width: $X = \lambda D / d$.
• Brewster's Law: $\mu = \tan i_p$.
• Malus' Law: $I = I_0 \cos^2\theta$.

Solved Example Q. YDSE: slits 1mm apart, screen 1m away, $\lambda = 5000$Å. Find fringe width.

$X = \frac{5 \times 10^{-7} \times 1}{10^{-3}}$

$X = 5 \times 10^{-4}$ m

Ans: 0.5 mm

8. Electrostatics

Key Formulas

• Force: $F = \frac{1}{4\pi\epsilon_0} \frac{q_1q_2}{r^2}$.
• Field: $E = F/q$.
• Potential: $V = W/q$.
• Capacitor Energy: $U = \frac{1}{2}CV^2$.

Solved Example Q. Capacitor 4 $\mu$F connected to 200V. Find energy.

$U = \frac{1}{2} \times 4 \times 10^{-6} \times (200)^2$

$U = 2 \times 10^{-6} \times 40000$

Ans: 0.08 Joules

9. Current Electricity

Key Formulas

• Ohm's Law: $V = IR$.
• Kirchhoff's Laws: $\sum I=0, \sum V=0$.
• Wheatstone: $R_1/R_2 = R_3/R_4$.
• Potentiometer: $E_1/E_2 = L_1/L_2$.

Solved Example Q. Wire of $10\Omega$ is stretched to double its length. New resistance?

Volume constant shortcut: $R_{new} = n^2 R_{old}$.

$R_{new} = (2)^2 \times 10 = 40$.

Ans: 40 $\Omega$

10. Magnetic Effects

Key Formulas

• Biot-Savart: $dB = \frac{\mu_0 I dl \sin\theta}{4\pi r^2}$.
• Ampere's Law: $\oint B \cdot dl = \mu_0 I$.
• Solenoid Field: $B = \mu_0 n I$.
• Lorentz Force: $F = q(v \times B)$.

Solved Example Q. Solenoid length 50cm, 100 turns, 2A current. Find B at center.

$n = N/L = 100/0.5 = 200$ turns/m.

$B = 4\pi \times 10^{-7} \times 200 \times 2$

Ans: $5.02 \times 10^{-4}$ T

11. Magnetism

Key Formulas

• Orbital Moment: $m_{orb} = \frac{e v r}{2}$.
• Axial Field: $B_a = \frac{\mu_0 2M}{4\pi r^3}$.
• Equatorial: $B_{eq} = \frac{\mu_0 M}{4\pi r^3}$.
• Torque: $\tau = m B \sin\theta$.

Solved Example Q. $M=5$ Am$^2$. Find axial B at 20 cm.

$B = 10^{-7} \times \frac{2 \times 5}{(0.2)^3}$

$B = 10^{-7} \times 1250$

Ans: $1.25 \times 10^{-4}$ T

12. Electromagnetic Induction

Key Formulas

• Flux: $\phi = B A \cos\theta$.
• Faraday's Law: $e = -d\phi/dt$.
• Self Induction: $e = -L(dI/dt)$.
• Transformer: $E_s/E_p = N_s/N_p$.

Solved Example Q. Flux changes from 5 Wb to 2 Wb in 0.1s. Find EMF.

$|e| = |(2-5)/0.1|$

$e = 3 / 0.1$

Ans: 30 Volts

13. AC Circuits

Key Formulas

• RMS: $I_{rms} = I_0 / \sqrt{2}$.
• Inductive Reactance: $X_L = \omega L$.
• Capacitive Reactance: $X_C = 1/\omega C$.
• Impedance: $Z = \sqrt{R^2 + (X_L - X_C)^2}$.

Solved Example Q. $V = 100 \sin(100 \pi t)$, $R=50\Omega$. Find $I_{rms}$.

$V_0 = 100 \Rightarrow I_0 = 100/50 = 2$A.

$I_{rms} = 2 / 1.414$

Ans: 1.414 A

[Image of LCR circuit diagram]

14. Dual Nature of Radiation

Key Formulas

• Einstein Eq: $E = \phi_0 + K_{max}$.
• Momentum: $p = h/\lambda$.
• Cut-off wavelength: $\lambda_0 = hc/\phi_0$.

Solved Example Q. Work function 2.5 eV. Find threshold wavelength.

$\lambda_0 = \frac{12400}{2.5}$ (Shortcut in Å)

Or use basic units: $\lambda_0 = \frac{hc}{2.5 \times 1.6 \times 10^{-19}}$

Ans: ~4960 Å

15. Structure of Atoms

Key Formulas

• Radius: $r_n \propto n^2$.
• Energy: $E_n = -13.6/n^2$ eV.
• Rydberg: $1/\lambda = R(1/n^2 - 1/m^2)$.
• Decay: $N = N_0(1/2)^n$.

Solved Example Q. Half-life 3 days. Fraction remaining after 9 days?

$n = t/T = 9/3 = 3$ half lives.

Remains $= (1/2)^3$

Ans: 1/8

16. Semiconductors

Key Formulas

• Current Gain: $\beta = I_c / I_b$.
• Relation: $\alpha = \beta / (1+\beta)$.
• Logic Gates: NAND ($Y=\overline{A \cdot B}$), NOR ($Y=\overline{A+B}$).

Solved Example Q. NAND Gate inputs A=1, B=1. Output?

AND is $1 \times 1 = 1$.

NAND inverts it to 0.

Ans: 0 (Low)

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