Physics Equation Sheet and Moments of Inertia Reference

Physics Equation Sheet

Fundamental Constants and Geometry

• Pi (\(\pi\)): \(\pi = 3.14\)

• Circumference of a circle: \(C = 2\pi r\)

• Area of a circle: \(A = \pi r^2\)

• Surface area of a sphere: \(A = 4\pi r^2\)

• 1 revolution: \(1 \text{ rev} = 2\pi = 360^\circ\)

• Acceleration due to gravity: \(g = 9.8\ \mathrm{m/s^2}\)

• Universal gravitational constant: \(G = 6.67 \times 10^{-11}\ \mathrm{N\,m^2/kg^2}\)

• Avogadro's number: \(N_A = 6.02 \times 10^{23}\)

• Boltmann constant: \(k_B = 1.38 \times 10^{-23}\ \mathrm{J/K}\)

Kinematics

• Quadratic formula: For \(0 = ax^2 + bx + c\), \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

• Time interval: \(\Delta t = t - t_0\)

• Average velocity: \(v_{x,\text{avg}} = \frac{\Delta x}{\Delta t} = \frac{x_f - x_0}{t_f - t_0}\)

• Average acceleration: \(a_{x,\text{avg}} = \frac{\Delta v_x}{\Delta t} = \frac{v_{x,f} - v_{x,0}}{t_f - t_0}\)

• Kinematic equations (constant acceleration):

  • \(v_x = v_{x,0} + a_x \Delta t\)

  • \(v_x^2 = v_{x,0}^2 + 2a_x(x - x_0)\)

  • \(x = x_0 + v_{x,0}\Delta t + \frac{1}{2}a_x(\Delta t)^2\)

  • \(v_{x,\text{avg}} = \frac{v_{x,0} + v_{x,f}}{2}\)

• Range equation (projectile motion): \(x = \frac{v_0^2 \sin(2\theta)}{g}\)

Rotational Kinematics

• Average angular velocity: \(\omega_{\text{avg}} = \frac{\Delta \theta}{\Delta t} = \frac{\theta_f - \theta_0}{t_f - t_0}\)

• Average angular acceleration: \(\alpha_{\text{avg}} = \frac{\Delta \omega}{\Delta t} = \frac{\omega_f - \omega_0}{t_f - t_0}\)

• Rotational kinematic equations (constant angular acceleration):

  • \(\omega = \omega_0 + \alpha \Delta t\)

  • \(\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)\)

  • \(\theta = \theta_0 + \omega_0 \Delta t + \frac{1}{2}\alpha (\Delta t)^2\)

  • \(\omega_{\text{avg}} = \frac{\omega_0 + \omega_f}{2}\)

Newtonian Mechanics

• Newton's Second Law (linear): \(F_{\text{net},x} = ma_x\)

• Newton's Second Law (vertical): \(F_{\text{net},y} = ma_y\)

• Weight: \(F_g = mg\)

• Universal law of gravitation: \(F_g = \frac{Gm_1m_2}{r^2}\)

• Friction (static): \(f_s = \mu_s F_N\)

• Friction (kinetic): \(f_k = \mu_k F_N\)

• Centripetal acceleration: \(a_c = \frac{v^2}{r}\)

• Speed in circular motion: \(v = \frac{2\pi r}{T}\)

• Period and frequency: \(T = \frac{1}{f}\)

Work, Energy, and Power

• Work: \(W = Fd \cos \theta\)

• Change in energy: \(\Delta E = W_{\text{net}}\)

• Kinetic energy (linear): \(K = \frac{1}{2}mv^2\)

• Kinetic energy (rotational): \(K = \frac{1}{2}I\omega^2\)

• Potential energy (gravitational): \(U = mgy\)

• Potential energy (spring): \(U = \frac{1}{2}kx^2\)

• Power (average): \(P_{\text{avg}} = \frac{W}{\Delta t}\)

Linear and Angular Momentum

• Linear momentum: \(p_x = mv_x\)

• Impulse: \(\Delta p_x = F_x \Delta t\)

• Angular momentum: \(L = I\omega\)

• Change in angular momentum: \(\Delta L = \tau \Delta t\)

Rotational Dynamics

• Torque: \(\tau = I\alpha\)

• Torque (force): \(\tau = rF \sin \theta = r_\perp F = rF_\perp\)

Oscillations and Waves

• Simple harmonic motion (spring): \(F = -kx\)

• Angular frequency: \(\omega = 2\pi f\)

• Displacement: \(x = A \cos(\omega t)\)

• Velocity: \(v_x = -\omega A \sin(\omega t)\)

• Acceleration: \(a_x = -\omega^2 A \cos(\omega t)\)

• Period (pendulum): \(T = 2\pi \sqrt{\frac{L}{g}}\)

• Period (spring): \(T = 2\pi \sqrt{\frac{m}{k}}\)

Mechanical Waves

• Wave equation: \(y = A \cos\left(2\pi \frac{t}{T} \pm 2\pi \frac{x}{\lambda}\right)\)

• Wave speed: \(v = \lambda f\)

• Speed on a string: \(v = \sqrt{\frac{F_T}{\mu}}\)

Fluids and Thermodynamics

• Pressure: \(P = \frac{F}{A}\)

• Continuity equation: \(Q = vA\)

• Bernoulli's equation: \(P_1 + \frac{1}{2}\rho v_1^2 + \rho g y_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g y_2\)

• Ideal gas law: \(PV = nRT\)

• Specific heat: \(Q = mc\Delta T\)

• Latent heat: \(Q = mL\)

• First law of thermodynamics: \(\Delta U_{int} = Q + W\)

• Second law of thermodynamics (entropy): \(\Delta S = \frac{Q_{rev}}{T}\)

Electricity and Magnetism

• Current density: \(J = \frac{I}{A}\)

• Intensity (sound): \(I = \frac{P}{A}\)

• Sound level (decibels): \(\beta = 10 \log_{10}\left(\frac{I}{I_0}\right)\)

Mathematical Tools

• Right triangle relations: \(\sin \theta = \frac{a}{c}, \cos \theta = \frac{b}{c}, \tan \theta = \frac{a}{b}\)

• Pythagorean theorem: \(a^2 + b^2 = c^2\)

Moments of Inertia for Common Objects

Reference Table: Moments of Inertia

The moment of inertia quantifies an object's resistance to changes in rotational motion about a specified axis. It depends on the mass distribution relative to the axis of rotation. Below is a summary table for common shapes:

Object	Location of Axis	Moment of Inertia
Thin hoop, radius R	Through center	$I = MR^2$
Thin hoop, radius R	Through central diameter	$I = \frac{1}{2}MR^2$
Solid cylinder, radius R	Through center	$I = \frac{1}{2}MR^2$
Hollow cylinder, inner radius $R_1$, outer radius $R_2$	Through center	$I = \frac{1}{2}M(R_1^2 + R_2^2)$
Uniform sphere, radius R	Through center	$I = \frac{2}{5}MR^2$
Long uniform rod, length L	Through center	$I = \frac{1}{12}ML^2$
Long uniform rod, length L	Through end	$I = \frac{1}{3}ML^2$
Rectangular thin plate, length a, width b	Through center, axis perpendicular to plate	$I = \frac{1}{12}M(a^2 + b^2)$

• Application: Use these formulas to calculate rotational kinetic energy, angular acceleration, and analyze rotational dynamics problems.

Additional info: The table above is essential for solving problems in rotational motion, especially when applying Newton's second law for rotation (\(\tau = I\alpha\)) and calculating the energy of rotating bodies.