PGT Physics MP Varg 1

Physics Formulas for MP Varg 1

1. Physical World and Measurement

• \( \Delta a = |a_{\text{measured}} - a_{\text{true}}| \)
• \( \frac{\Delta a}{a} \)
• \( \frac{\Delta a}{a} \times 100\% \)
• \( \Delta Z = \Delta A + \Delta B \)
• \( \frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B} \)
• \( \frac{\Delta Z}{Z} = n \frac{\Delta A}{A} \) (for \( Z = A^n \))

2. Kinematics

• \( v = u + at \)
• \( s = ut + \frac{1}{2}at^2 \)
• \( v^2 = u^2 + 2as \)
• \( s_n = u + \frac{a}{2}(2n-1) \)
• \( \vec{v}_{AB} = \vec{v}_A - \vec{v}_B \)
• \( T = \frac{2u \sin\theta}{g} \)
• \( R = \frac{u^2 \sin2\theta}{g} \)
• \( H = \frac{u^2 \sin^2\theta}{2g} \)

3. Laws of Motion

• \( F = 0 \implies a = 0 \)
• \( \vec{F} = m\vec{a} \)
• \( \vec{F}_{AB} = -\vec{F}_{BA} \)
• \( f_s \leq \mu_s N \)
• \( f_k = \mu_k N \)
• \( F = \frac{mv^2}{r} = m\omega^2 r \)
• \( \omega = \frac{v}{r} \)

4. Work, Energy, and Power

• \( W = \vec{F} \cdot \vec{s} = Fs \cos\theta \)
• \( KE = \frac{1}{2}mv^2 \)
• \( PE = mgh \)
• \( PE = \frac{1}{2}kx^2 \)
• \( W = \Delta KE \)
• \( KE_i + PE_i = KE_f + PE_f \)
• \( P = \frac{W}{t} = \vec{F} \cdot \vec{v} \)

5. Motion of System of Particles and Rigid Body

• \( \vec{r}_{cm} = \frac{\sum m_i \vec{r}_i}{\sum m_i} \)
• \( \vec{v}_{cm} = \frac{\sum m_i \vec{v}_i}{\sum m_i} \)
• \( \vec{\tau} = \vec{r} \times \vec{F} \)
• \( \vec{L} = \vec{r} \times \vec{p} \)
• \( I = \sum m_i r_i^2 \)
• \( I = \frac{2}{5}MR^2 \)
• \( I = \frac{2}{3}MR^2 \)
• \( I = \frac{1}{2}MR^2 \)
• \( I = \frac{1}{12}ML^2 \)
• \( \omega = \omega_0 + \alpha t \)
• \( \theta = \omega_0 t + \frac{1}{2}\alpha t^2 \)
• \( \omega^2 = \omega_0^2 + 2\alpha \theta \)
• \( \tau = I \alpha \)

6. Gravitation

• \( F = \frac{G M_1 M_2}{r^2} \)
• \( U = -\frac{G M_1 M_2}{r} \)
• \( g = \frac{GM}{R^2} \)
• \( g_h = g \left(1 - \frac{2h}{R}\right) \)
• \( g_d = g \left(1 - \frac{d}{R}\right) \)
• \( v_e = \sqrt{\frac{2GM}{R}} \)
• \( v_o = \sqrt{\frac{GM}{r}} \)
• \( T = 2\pi \sqrt{\frac{r^3}{GM}} \)

7. Properties of Bulk Matter

• \( \sigma = \frac{F}{A} \)
• \( \epsilon = \frac{\Delta L}{L} \text{ or } \frac{\Delta V}{V} \)
• \( Y = \frac{\text{Stress}}{\text{Strain}} \)
• \( K = -\frac{\Delta P}{\Delta V / V} \)
• \( G = \frac{\text{Shear Stress}}{\text{Shear Strain}} \)
• \( T = \frac{F}{l} \)
• \( F = \eta A \frac{dv}{dx} \)
• \( P + \rho g h + \frac{1}{2} \rho v^2 = \text{constant} \)
• \( Q = \frac{\pi P r^4}{8 \eta l} \)

8. Thermodynamics

• \( \Delta Q = \Delta U + W \)
• \( W = nRT \ln\left(\frac{V_f}{V_i}\right) \)
• \( W = \frac{P_i V_i - P_f V_f}{\gamma - 1} \)
• \( \gamma = \frac{C_p}{C_v} \)
• \( PV^\gamma = \text{constant} \)
• \( TV^{\gamma-1} = \text{constant} \)
• \( \eta = 1 - \frac{T_2}{T_1} \)

9. Oscillations and Waves

• \( x = A \sin(\omega t + \phi) \)
• \( v = A \omega \cos(\omega t + \phi) \)
• \( a = - \omega^2 x \)
• \( T = 2\pi \sqrt{\frac{m}{k}} \)
• \( T = 2\pi \sqrt{\frac{l}{g}} \)
• \( \omega = \sqrt{\frac{k}{m}} \)
• \( v = f \lambda \)
• \( v = \sqrt{\frac{T}{\mu}} \)
• \( v = \sqrt{\frac{B}{\rho}} \)
• \( f' = f \left( \frac{v \pm v_o}{v \mp v_s} \right) \)

10. Electrostatics

• \( F = \frac{k q_1 q_2}{r^2} \), \( k = \frac{1}{4\pi \epsilon_0} \)
• \( \vec{E} = \frac{k q}{r^2} \hat{r} \)
• \( V = \frac{k q}{r} \)
• \( U = \frac{k q_1 q_2}{r} \)
• \( C = \frac{Q}{V} \)
• \( C = \frac{\epsilon_0 A}{d} \)
• \( U = \frac{1}{2} C V^2 = \frac{Q^2}{2C} \)
• \( \Phi_E = \frac{q}{\epsilon_0} \)

11. Current Electricity

• \( V = IR \)
• \( R = \frac{\rho l}{A} \)
• \( R_{\text{eq}} = R_1 + R_2 + \dots \)
• \( \frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots \)
• \( P = VI = I^2 R = \frac{V^2}{R} \)
• \( \sum I_{\text{in}} = \sum I_{\text{out}} \)
• \( \sum V = 0 \)
• \( \frac{R_1}{R_2} = \frac{R_3}{R_4} \)

12. Magnetic Effects of Current and Magnetism

• \( d\vec{B} = \frac{\mu_0 I}{4\pi} \frac{d\vec{l} \times \hat{r}}{r^2} \)
• \( B = \frac{\mu_0 I}{2\pi r} \)
• \( B = \frac{\mu_0 I}{2R} \)
• \( B = \mu_0 n I \)
• \( \vec{F} = I (\vec{l} \times \vec{B}) \)
• \( \vec{F} = q (\vec{v} \times \vec{B}) + q \vec{E} \)
• \( \vec{m} = I \vec{A} \)
• \( \vec{\tau} = \vec{m} \times \vec{B} \)

13. Electromagnetic Induction and Alternating Currents

• \( \mathcal{E} = -\frac{d\Phi_B}{dt} \), \( \Phi_B = \vec{B} \cdot \vec{A} \)
• \( \mathcal{E} = -L \frac{di}{dt} \)
• \( \mathcal{E}_2 = -M \frac{di_1}{dt} \)
• \( V = V_0 \sin(\omega t) \), \( I = I_0 \sin(\omega t) \)
• \( V_{\text{rms}} = \frac{V_0}{\sqrt{2}} \), \( I_{\text{rms}} = \frac{I_0}{\sqrt{2}} \)
• \( X_L = \omega L \)
• \( X_C = \frac{1}{\omega C} \)
• \( Z = \sqrt{R^2 + (X_L - X_C)^2} \)
• \( P_{\text{avg}} = V_{\text{rms}} I_{\text{rms}} \cos\phi \)

14. Optics

• \( \angle i = \angle r \)
• \( \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \)
• \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \)
• \( \frac{1}{f} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \)
• \( m = \frac{h_i}{h_o} = \frac{v}{u} \)
• \( n_1 \sin\theta_1 = n_2 \sin\theta_2 \)
• \( \theta_c = \sin^{-1}\left(\frac{n_2}{n_1}\right) \)
• \( \beta = \frac{\lambda D}{d} \)
• \( d \sin\theta = n\lambda \)

15. Modern Physics

• \( KE_{\text{max}} = h\nu - \phi \)
• \( \lambda = \frac{h}{p} \)
• \( r_n = \frac{n^2 h^2}{4\pi^2 m k Z e^2} \)
• \( E_n = -\frac{13.6 Z^2}{n^2} \, \text{eV} \)
• \( N = N_0 e^{-\lambda t} \)
• \( T_{1/2} = \frac{\ln 2}{\lambda} \)
• \( E = mc^2 \)

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Practice Quizzes (Chapter Wise)

Select a Chapter 1. Physical World and Measurement 2. Kinematics 3. Laws of Motion 4. Work, Energy, and Power 5. Motion of System of Particles and Rigid Body 6. Gravitation 7. Properties of Bulk Matter 8. Thermodynamics 9. Oscillations and Waves 10. Electrostatics 11. Current Electricity 12. Magnetic Effects of Current and Magnetism 13. Electromagnetic Induction and Alternating Currents 14. Optics 15. Modern Physics