PHYS1425 Formula Sheet: Mechanics, Energy, and Fluids

Mathematical Tools

Quadratic Equation

The quadratic equation is used to solve for x in equations of the form ax^2 + bx + c = 0.

• General Solution:

$x = \frac{ -b \pm \sqrt{ b^2 - 4ac } }{ 2a }$

Chapter 2: Kinematics in One Dimension

1D Motion with Constant Acceleration

Describes the motion of objects moving in a straight line with constant acceleration.

• Position as a function of time:

$x(t) = x_0 + v_0 t + \frac{1}{2} a t^2$

• Velocity as a function of time:

$v(t) = v_0 + a t$

• Velocity-position relation:

$v^2(t) - v_0^2 = 2a [ x(t) - x_0 ]$

• Average velocity:

$\bar{v} = \frac{ v(t) + v_0 }{ 2 }$

• Free fall acceleration: $g = 9.8\ \mathrm{m/s^2}$ (downward, near Earth's surface)

Chapter 3: Kinematics in Two or Three Dimensions; Vectors

Vector Representation

• Component form: $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$

• Magnitude: $|\vec{A}| = \sqrt{ A_x^2 + A_y^2 + A_z^2 }$

2D Projectile Motion

• Horizontal position: $x(t) = x_0 + v_{0,x} t$

• Vertical position: $y(t) = y_0 + v_{0,y} t - \frac{1}{2} g t^2$

• Horizontal velocity: $v_x(t) = v_{0,x}$

• Vertical velocity: $v_y(t) = v_{0,y} - g t$

• Range (for same launch and landing height): $R = \frac{ v_0^2 \sin(2\theta_0) }{ g }$

Chapter 4: Dynamics: Newton's Laws of Motion

Newton's Second Law

• Relation between force and acceleration:

$\vec{a} = \frac{ \vec{F} }{ m }$

• SI unit of force: Newton (N)

Chapter 5: Using Newton's Laws: Friction, Circular Motion, Drag Forces

Friction

• Kinetic friction: $f_k = \mu_k N$

• Static friction (maximum): $f_s \leq f_{s,\text{max}} = \mu_s N$

Uniform Circular Motion

• Centripetal acceleration: $a_{cp} = \frac{ v^2 }{ R }$

• Period: $T = \frac{ 2\pi R }{ v }$

Drag Force

• Linear drag: $f_D = -b v$

Chapter 6: Gravitation and Newton's Synthesis

Newton's Law of Universal Gravitation

• Gravitational force: $F = G \frac{ m_1 m_2 }{ r^2 }$

• Gravitational constant: $G = 6.67 \times 10^{-11}\ \mathrm{N \cdot m^2 / kg^2}$

Satellite Motion

• Orbital speed: $v = \sqrt{ \frac{ GM }{ r } }$

• Orbital period: $T = \frac{ 2\pi r^{3/2} }{ \sqrt{ GM } }$

Kepler's Third Law

• Relation between periods and semi-major axes: $\left( \frac{ T_1 }{ T_2 } \right)^2 = \left( \frac{ s_1 }{ s_2 } \right)^3$

Chapter 7: Work and Energy

Dot Product of Vectors

• Definition: $\vec{A} \cdot \vec{B} \equiv A_x B_x + A_y B_y + A_z B_z = |\vec{A}| |\vec{B}| \cos \theta$

Work

• General definition: $W = \int_A^B \vec{F} \cdot d\vec{l}$

• Constant force: $W = \vec{F} \cdot \vec{l} = |\vec{F}| |\vec{l}| \cos \theta$

• Work by a spring: $W = \frac{1}{2} k ( x_A^2 - x_B^2 )$

• Work by gravity (general): $W = G m M \left( \frac{1}{ r_B } - \frac{1}{ r_A } \right)$

• Work by gravity (near Earth): $W = m g ( h_A - h_B )$

Kinetic Energy

• Definition: $K = \frac{1}{2} m v^2$

• SI unit: Joule (J)

Work-Energy Principle

• Statement: $\sum W = K_f - K_i$

Chapter 8: Conservation of Energy

Potential Energy

• Definition: $\Delta U = -W_{\text{cons}}$

• Elastic potential energy: $U = \frac{1}{2} k x^2$

• Gravitational potential energy (general): $U = - \frac{ G m M }{ r }$

• Gravitational potential energy (near Earth): $U = m g h$

Conservation of Mechanical Energy

• Without friction: $K_i + U_i = K_f + U_f$

• With friction: $K_i + U_i = K_f + U_f + Q$ where $Q = | W_{fr} |$

Escape Velocity

• Formula: $v_{esc} = \sqrt{ \frac{ 2 G M }{ R } }$

• For Earth (from surface): $v_{esc} = 11.2\ \mathrm{km/s}$

Chapter 9: Linear Momentum

Impulse and Momentum

• Impulse: $\vec{I} \equiv \int \vec{F} dt = \Delta \vec{p} = \vec{p}_f - \vec{p}_i$

• Linear momentum: $\vec{p} = m \vec{v}$

• Newton's law (momentum form): $\vec{F} = \frac{ d \vec{p} }{ dt }$

• Conservation of momentum (isolated system): $\vec{p}_i = \vec{p}_f$

1D Collisions

• Elastic collision (A moving, B at rest):

$v'_A = \frac{ m_A - m_B }{ m_A + m_B } v_A$

$v'_B = \frac{ 2 m_A }{ m_A + m_B } v_A$

• Elastic collision (both moving):

$v'_A = \frac{ m_A - m_B }{ m_A + m_B } v_A + \frac{ 2 m_B }{ m_A + m_B } v_B$

$v'_B = \frac{ 2 m_A }{ m_A + m_B } ( v_A - v_B ) + v_B = \frac{ m_B - m_A }{ m_A + m_B } v_B + \frac{ 2 m_A }{ m_A + m_B } v_A$

• Completely inelastic collision (both moving):

$v_f = \frac{ m_A v_A + m_B v_B }{ m_A + m_B }$

Center of Mass

• Discrete system: $x_{CM} = \frac{ \sum_i m_i x_i }{ \sum m_i }$, $y_{CM} = \frac{ \sum_i m_i y_i }{ \sum m_i }$

• Continuous distribution: $x_{CM} = \frac{1}{m} \int x dm$, $y_{CM} = \frac{1}{m} \int y dm$

• Newton's 2nd law for system: $\vec{a}_{CM} = \frac{ \sum \vec{F} }{ M }$

Chapter 10: Rotational Motion

Angular Motion

• Angular position, velocity, acceleration: $\theta$, $\omega = \frac{ d\theta }{ dt }$, $\alpha = \frac{ d\omega }{ dt }$

• One full turn: $2\pi$ radians

• Uniform circular motion: $|\omega| = \frac{ 2\pi }{ T }$

• Angular displacement (constant acceleration): $\theta(t) = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$

• Angular velocity (constant acceleration): $\omega(t) = \omega_0 + \alpha t$

• Angular velocity-displacement relation: $\omega^2(t) - \omega_0^2 = 2\alpha [ \theta(t) - \theta_0 ]$

• Average angular velocity: $\bar{\omega} = \frac{ \omega(t) + \omega_0 }{ 2 }$

Relation Between Angular and Linear Quantities

• Tangential velocity: $v_t = \omega r$

• Tangential acceleration: $a_t = \alpha r$

• Centripetal acceleration: $a_{cp} = \frac{ v_t^2 }{ r } = \omega^2 r$

Torque

• Vector definition: $\vec{\tau} = \vec{r} \times \vec{F}$

• Magnitude: $\tau = r F \sin \theta = r F_t = r_L F$

Rotational Inertia (Moment of Inertia)

• Integral form: $I = \int r^2 dm$

• Discrete sum: $I = \sum_i m_i R_i^2$

• Parallel axis theorem: $I = I_{CM} + m h^2$

Newton's Second Law for Rotation

• For fixed axis: $\sum \vec{\tau} = I \vec{\alpha}$

Rolling Motion (No Slipping)

• Relation: $v_{CM} = \omega R$

Rotational Kinetic Energy

• Fixed axis: $K = \frac{1}{2} I \omega^2$

• Total (rolling): $K_{total} = \frac{1}{2} m v_{CM}^2 + \frac{1}{2} I_{CM} \omega_{CM}^2$

Chapter 11: Angular Momentum; General Rotation

Vector Cross Product

• Definition: $\vec{C} = \vec{A} \times \vec{B}$

• Components:

$C_x = A_y B_z - A_z B_y$

$C_y = A_z B_x - A_x B_z$

$C_z = A_x B_y - A_y B_x$

Angular Momentum

• General definition: $\vec{L} = \vec{r} \times \vec{p} = \vec{r} \times ( m \vec{v} )$

• For rotation about fixed axis: $L = I \omega$

Newton's Second Law for Rotation (Angular Momentum Form)

• General form: $\sum \vec{\tau} = \frac{ d \vec{L} }{ dt }$

Conservation of Angular Momentum

• Statement: $\vec{L}_i = \vec{L}_f$ (if net external torque is zero)

Chapter 12: Static Equilibrium; Elasticity and Fracture

Conditions for Static Equilibrium

• Translational equilibrium: $\sum \vec{F} = 0$

• Rotational equilibrium: $\sum \tau = 0$

Elastic Moduli

• Young's modulus: $E = \frac{ F / A }{ \Delta l / l_0 }$

• Shear modulus: $G = \frac{ F / A }{ \Delta l / l_0 }$

• Bulk modulus: $B = - \frac{ \Delta P }{ \Delta V / V_0 }$

Safety Factor

• Definition: Ratio of strength over expected load or expected stress

Chapter 13: Fluids

Density

• Definition: $\rho = \frac{ m }{ V }$

• Density of water: $\rho = 1.0 \times 10^3\ \mathrm{kg/m^3} = 1.0\ \mathrm{g/cm^3}$

Pressure in Fluids

• Hydrostatic pressure: $P = \rho g h$

• Atmospheric pressure: $1\ \mathrm{atm} = 1.013 \times 10^5\ \mathrm{Pa} = 14.6\ \mathrm{psi}$

• Gauge pressure: $P_G = P - P_{atm}$

Example Application: A ball is thrown horizontally from a height of 20 m with an initial speed of 10 m/s. Using the projectile motion equations, you can calculate the time to hit the ground and the horizontal distance traveled. Additional info: This formula sheet covers the core equations and definitions for introductory mechanics, energy, and fluids, suitable for exam reference in a college physics course.