Formula Sheet: Translational Motion and Calculations

Table of Contents
1. Kinematics: Linear Motion
2. Projectile Motion
3. Newton's Laws of Motion
4. Forces and Applications
5. Circular Motion
6. Work, Energy, and Power
7. Momentum and Collisions
8. Rotational Motion (Translational Analogs)
9. Gravity and Orbital Motion
10. Simple Harmonic Motion
11. Fluids at Rest (Hydrostatics)
12. Fluids in Motion (Fluid Dynamics)
13. Additional Important Relationships
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Kinematics: Linear Motion

Fundamental Kinematic Variables

• Position (x): Location of an object in space; SI unit: meters (m)
• Displacement (Δx): Change in position; vector quantity; SI unit: meters (m) \[\Delta x = x_f - x_i\] where \(x_f\) = final position, \(x_i\) = initial position
• Distance: Total path length traveled; scalar quantity; SI unit: meters (m)
• Velocity (v): Rate of change of position; vector quantity; SI unit: m/s \[v = \frac{\Delta x}{\Delta t}\]
• Speed: Magnitude of velocity; scalar quantity; SI unit: m/s
• Acceleration (a): Rate of change of velocity; vector quantity; SI unit: m/s² \[a = \frac{\Delta v}{\Delta t}\]

Kinematic Equations for Constant Acceleration

Valid only when acceleration is constant

• Equation 1 (Velocity-Time): \[v_f = v_i + at\] where \(v_f\) = final velocity, \(v_i\) = initial velocity, \(a\) = acceleration, \(t\) = time
• Equation 2 (Position-Time): \[x = x_i + v_i t + \frac{1}{2}at^2\] where \(x\) = final position, \(x_i\) = initial position
• Equation 3 (Velocity-Position): \[v_f^2 = v_i^2 + 2a\Delta x\] Note: Use when time is unknown or not needed
• Equation 4 (Average Velocity): \[v_{avg} = \frac{v_i + v_f}{2}\] Valid only for constant acceleration
• Equation 5 (Displacement using Average Velocity): \[\Delta x = v_{avg} \cdot t = \frac{(v_i + v_f)}{2} \cdot t\]

Free Fall Motion

Special case of constant acceleration where a = g

• Acceleration due to gravity (g): \[g = 9.8 \text{ m/s}^2 \approx 10 \text{ m/s}^2\] Direction: Always points downward toward Earth's center
• Free fall equations: Replace \(a\) with \(g\) (or -g depending on coordinate system) \[v_f = v_i + gt\] \[y = y_i + v_i t + \frac{1}{2}gt^2\] \[v_f^2 = v_i^2 + 2g\Delta y\] Convention: Often use y for vertical position; choose + or - for g based on coordinate system
• At maximum height: \(v = 0\)
• Time to reach maximum height: \[t_{max} = \frac{v_i}{g}\] (when thrown upward with initial velocity \(v_i\))
• Maximum height: \[h_{max} = \frac{v_i^2}{2g}\]
• Symmetry property: Time up = Time down; Speed going up at height h = Speed coming down at height h

Projectile Motion

Fundamental Principles

• Independence of motion: Horizontal and vertical motions are independent
• Horizontal motion: Constant velocity (no acceleration, neglecting air resistance)
• Vertical motion: Constant acceleration due to gravity
• Time of flight: Same for both horizontal and vertical components

Component Analysis

Initial Velocity Components (launch angle θ from horizontal):

• Horizontal component: \[v_{ix} = v_i \cos\theta\]
• Vertical component: \[v_{iy} = v_i \sin\theta\]

Horizontal Motion Equations

• Horizontal velocity: \(v_x = v_{ix} = \text{constant}\)
• Horizontal displacement: \[x = v_{ix} \cdot t = v_i \cos\theta \cdot t\]

Vertical Motion Equations

• Vertical velocity: \[v_y = v_{iy} - gt = v_i \sin\theta - gt\]
• Vertical displacement: \[y = v_{iy} t - \frac{1}{2}gt^2 = v_i \sin\theta \cdot t - \frac{1}{2}gt^2\]
• Vertical velocity-position relation: \[v_y^2 = v_{iy}^2 - 2g\Delta y\]

Key Projectile Motion Formulas

• Time of flight (returns to same height): \[t_{total} = \frac{2v_i \sin\theta}{g}\]
• Maximum height: \[h_{max} = \frac{v_i^2 \sin^2\theta}{2g}\]
• Range (horizontal distance): \[R = \frac{v_i^2 \sin(2\theta)}{g}\] Maximum range occurs at θ = 45°
• Velocity magnitude at any time: \[v = \sqrt{v_x^2 + v_y^2}\]
• Angle of velocity at any time: \[\tan\phi = \frac{v_y}{v_x}\] where φ is measured from horizontal

Newton's Laws of Motion

Newton's First Law (Law of Inertia)

• Statement: An object at rest stays at rest, and an object in motion stays in motion with constant velocity, unless acted upon by a net external force
• Mathematical expression: If \(\sum \vec{F} = 0\), then \(\vec{v} = \text{constant}\)
• Inertia: Tendency of an object to resist changes in motion; proportional to mass

Newton's Second Law

• Fundamental form: \[\sum \vec{F} = m\vec{a}\] where \(\sum \vec{F}\) = net force (N), \(m\) = mass (kg), \(\vec{a}\) = acceleration (m/s²)
• Component form: \[\sum F_x = ma_x\] \[\sum F_y = ma_y\]
• Alternative form (momentum): \[\sum \vec{F} = \frac{d\vec{p}}{dt}\] where \(\vec{p}\) = momentum
• Weight: \[W = mg\] where \(W\) = weight (N), \(g\) = 9.8 m/s²

Newton's Third Law

• Statement: For every action force, there is an equal and opposite reaction force
• Mathematical expression: \[\vec{F}_{AB} = -\vec{F}_{BA}\] where \(\vec{F}_{AB}\) is force on object A by object B
• Key point: Action-reaction pairs act on different objects

Forces and Applications

Friction

Static Friction (object at rest):

• Maximum static friction: \[f_s \leq \mu_s N\] where \(f_s\) = static friction force, \(\mu_s\) = coefficient of static friction, \(N\) = normal force
• At the threshold of motion: \(f_s = \mu_s N\)

Kinetic Friction (object in motion):

• Kinetic friction force: \[f_k = \mu_k N\] where \(f_k\) = kinetic friction force, \(\mu_k\) = coefficient of kinetic friction
• Important: Generally \(\mu_k <>
• Direction: Always opposes motion

Inclined Plane

Forces on an inclined plane (angle θ from horizontal):

• Weight component parallel to incline: \[F_{\parallel} = mg\sin\theta\]
• Weight component perpendicular to incline: \[F_{\perp} = mg\cos\theta\]
• Normal force (no acceleration perpendicular to surface): \[N = mg\cos\theta\]
• Friction force on incline: \[f = \mu N = \mu mg\cos\theta\]
• Acceleration down frictionless incline: \[a = g\sin\theta\]
• Acceleration with friction: \[a = g(\sin\theta - \mu_k\cos\theta)\] (down the incline)

Tension

• Tension in rope/string: Force transmitted through rope; pulls equally on objects at both ends
• Massless rope: Tension is same throughout the rope
• Rope with mass: Tension varies along length
• Over massless, frictionless pulley: Tension is same on both sides

Normal Force

• Definition: Perpendicular contact force exerted by a surface
• On horizontal surface (no vertical acceleration): \(N = mg\)
• In elevator accelerating upward: \(N = m(g + a)\)
• In elevator accelerating downward: \(N = m(g - a)\)
• On incline: \(N = mg\cos\theta\)

Circular Motion

Uniform Circular Motion

Constant speed in circular path

• Centripetal acceleration: \[a_c = \frac{v^2}{r}\] where \(v\) = tangential speed, \(r\) = radius of circular path
• Alternative form: \[a_c = \omega^2 r\] where \(\omega\) = angular velocity (rad/s)
• Centripetal force: \[F_c = \frac{mv^2}{r} = m\omega^2 r\] Direction: Always points toward center of circle
• Period (T): Time for one complete revolution \[T = \frac{2\pi r}{v}\]
• Frequency (f): Number of revolutions per unit time \[f = \frac{1}{T}\] SI unit: Hz (s^-1)
• Angular velocity: \[\omega = \frac{2\pi}{T} = 2\pi f = \frac{v}{r}\] SI unit: rad/s
• Relationship between linear and angular velocity: \[v = \omega r\]

Vertical Circular Motion

• At the top of circle: \[F_c = mg + N = \frac{mv^2}{r}\] (both weight and normal force point toward center)
• At the bottom of circle: \[F_c = N - mg = \frac{mv^2}{r}\] (normal force points toward center, weight points away)
• Minimum speed at top (for N ≥ 0): \[v_{min} = \sqrt{gr}\]

Banked Curves

For a banked curve with angle θ (no friction):

• Ideal banking angle: \[\tan\theta = \frac{v^2}{rg}\] where \(v\) = speed, \(r\) = radius of curve
• Ideal speed for given angle: \[v = \sqrt{rg\tan\theta}\]

Work, Energy, and Power

Work

• Work done by constant force: \[W = \vec{F} \cdot \vec{d} = Fd\cos\theta\] where \(F\) = force magnitude, \(d\) = displacement magnitude, \(\theta\) = angle between force and displacement
  SI unit: Joule (J) = N·m
• When force parallel to displacement: \(W = Fd\) (θ = 0°)
• When force perpendicular to displacement: \(W = 0\) (θ = 90°)
• When force opposite to displacement: \(W = -Fd\) (θ = 180°)
• Work by variable force: \[W = \int_{x_i}^{x_f} F \, dx\] (Equals area under F vs. x graph)
• Net work (work-energy theorem): \[W_{net} = \Delta KE = KE_f - KE_i\]

Kinetic Energy

• Kinetic energy: \[KE = \frac{1}{2}mv^2\] where \(m\) = mass (kg), \(v\) = speed (m/s)
  SI unit: Joule (J)
• Work-Energy Theorem: \[W_{net} = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2\]

Potential Energy

Gravitational Potential Energy:

• Near Earth's surface: \[PE_g = mgh\] where \(h\) = height above reference point, \(g\) = 9.8 m/s²
  SI unit: Joule (J)
• Change in gravitational PE: \[\Delta PE_g = mg\Delta h\]
• Work by gravity: \[W_g = -\Delta PE_g = -mg\Delta h\]

Elastic Potential Energy:

• Spring potential energy: \[PE_s = \frac{1}{2}kx^2\] where \(k\) = spring constant (N/m), \(x\) = displacement from equilibrium
  SI unit: Joule (J)
• Hooke's Law (restoring force): \[F_s = -kx\] Negative sign indicates force opposes displacement

Conservation of Energy

• Mechanical Energy: \[E = KE + PE\]
• Conservation of Mechanical Energy (no non-conservative forces): \[E_i = E_f\] \[KE_i + PE_i = KE_f + PE_f\] \[\frac{1}{2}mv_i^2 + mgh_i = \frac{1}{2}mv_f^2 + mgh_f\]
• With non-conservative forces (friction, air resistance): \[E_i = E_f + W_{nc}\] where \(W_{nc}\) = work done by non-conservative forces (energy lost)
• Energy dissipated by friction: \[E_{friction} = f_k \cdot d\] where \(d\) = distance traveled

Power

• Average power: \[P_{avg} = \frac{W}{\Delta t}\] where \(W\) = work done, \(\Delta t\) = time interval
  SI unit: Watt (W) = J/s
• Instantaneous power: \[P = \frac{dW}{dt} = \vec{F} \cdot \vec{v} = Fv\cos\theta\] where \(\theta\) = angle between force and velocity
• When force parallel to velocity: \[P = Fv\]
• Power in terms of energy: \[P = \frac{\Delta E}{\Delta t}\]
• Efficiency: \[\text{Efficiency} = \frac{P_{out}}{P_{in}} = \frac{W_{out}}{W_{in}}\] Often expressed as percentage

Momentum and Collisions

Linear Momentum

• Momentum: \[\vec{p} = m\vec{v}\] SI unit: kg·m/s
  Vector quantity in same direction as velocity
• Newton's Second Law (momentum form): \[\sum \vec{F} = \frac{d\vec{p}}{dt} = \frac{\Delta \vec{p}}{\Delta t}\]
• Impulse: \[\vec{J} = \sum \vec{F} \cdot \Delta t = \Delta \vec{p}\] SI unit: N·s = kg·m/s
• Impulse-Momentum Theorem: \[\vec{J} = \Delta \vec{p} = m\vec{v}_f - m\vec{v}_i\]
• For variable force: \[J = \int_{t_i}^{t_f} F \, dt\] (Area under F vs. t graph)

Conservation of Momentum

• Law of Conservation of Momentum: When net external force is zero: \[\vec{p}_{total,i} = \vec{p}_{total,f}\] \[\sum m_i \vec{v}_i = \sum m_f \vec{v}_f\]
• For two-object system: \[m_1 \vec{v}_{1i} + m_2 \vec{v}_{2i} = m_1 \vec{v}_{1f} + m_2 \vec{v}_{2f}\]
• Component form: Applies separately to x and y directions \[p_{x,i} = p_{x,f}\] \[p_{y,i} = p_{y,f}\]

Collisions

Elastic Collisions (both momentum and kinetic energy conserved):

• Momentum conservation: \[m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}\]
• Kinetic energy conservation: \[\frac{1}{2}m_1 v_{1i}^2 + \frac{1}{2}m_2 v_{2i}^2 = \frac{1}{2}m_1 v_{1f}^2 + \frac{1}{2}m_2 v_{2f}^2\]
• Relative velocity relationship: \[v_{1i} - v_{2i} = -(v_{1f} - v_{2f})\] (Relative velocity reverses in elastic collision)
• Equal mass elastic collision (m₁ = m₂, object 2 initially at rest):
  Object 1 stops: \(v_{1f} = 0\)
  Object 2 moves with initial velocity of object 1: \(v_{2f} = v_{1i}\)

Inelastic Collisions (only momentum conserved, KE not conserved):

• Momentum conservation: \[m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}\]
• Kinetic energy lost: \[\Delta KE = KE_f - KE_i <>

Perfectly Inelastic Collisions (objects stick together):

• Final velocity (objects move together): \[m_1 v_{1i} + m_2 v_{2i} = (m_1 + m_2)v_f\] \[v_f = \frac{m_1 v_{1i} + m_2 v_{2i}}{m_1 + m_2}\]
• Maximum kinetic energy lost (compared to other inelastic collisions)

Center of Mass

• Center of mass position (one dimension): \[x_{cm} = \frac{m_1 x_1 + m_2 x_2 + ... + m_n x_n}{m_1 + m_2 + ... + m_n} = \frac{\sum m_i x_i}{\sum m_i}\]
• Center of mass position (two dimensions): \[x_{cm} = \frac{\sum m_i x_i}{M_{total}}\] \[y_{cm} = \frac{\sum m_i y_i}{M_{total}}\]
• Velocity of center of mass: \[\vec{v}_{cm} = \frac{\sum m_i \vec{v}_i}{M_{total}} = \frac{\vec{p}_{total}}{M_{total}}\]
• For isolated system: Center of mass moves with constant velocity (or remains at rest)

Rotational Motion (Translational Analogs)

Rotational Kinematics

• Angular displacement (θ): SI unit: radians (rad) \[\Delta\theta = \theta_f - \theta_i\]
• Angular velocity (ω): \[\omega = \frac{d\theta}{dt} = \frac{\Delta\theta}{\Delta t}\] SI unit: rad/s
• Angular acceleration (α): \[\alpha = \frac{d\omega}{dt} = \frac{\Delta\omega}{\Delta t}\] SI unit: rad/s²
• Relationship to linear quantities:
  Arc length: \(s = r\theta\) (θ in radians)
  Linear velocity: \(v = r\omega\)
  Tangential acceleration: \(a_t = r\alpha\)
• Period and frequency: \[T = \frac{2\pi}{\omega}\] \[f = \frac{1}{T} = \frac{\omega}{2\pi}\]

Rotational Kinematic Equations (Constant Angular Acceleration)

• Angular velocity: \[\omega_f = \omega_i + \alpha t\]
• Angular displacement: \[\theta = \theta_i + \omega_i t + \frac{1}{2}\alpha t^2\]
• Velocity-displacement relation: \[\omega_f^2 = \omega_i^2 + 2\alpha\Delta\theta\]

Moment of Inertia

• Definition (rotational analog of mass): \[I = \sum m_i r_i^2\] For continuous object: \(I = \int r^2 \, dm\)
  SI unit: kg·m²
• Common moments of inertia (about center of mass):
  Point mass at distance r: \(I = mr^2\)
  Thin rod (axis through center, perpendicular): \(I = \frac{1}{12}ML^2\)
  Thin rod (axis through end): \(I = \frac{1}{3}ML^2\)
  Solid cylinder/disk (axis through center): \(I = \frac{1}{2}MR^2\)
  Hollow cylinder (axis through center): \(I = MR^2\)
  Solid sphere (axis through center): \(I = \frac{2}{5}MR^2\)
  Hollow sphere (axis through center): \(I = \frac{2}{3}MR^2\)
• Parallel Axis Theorem: \[I = I_{cm} + Md^2\] where \(I_{cm}\) = moment of inertia about center of mass, \(d\) = distance between parallel axes

Torque

• Torque (rotational analog of force): \[\tau = r F \sin\theta = rF_{\perp}\] where \(r\) = distance from axis, \(F\) = force, \(\theta\) = angle between r and F
  SI unit: N·m
• Alternative form: \[\tau = r_{\perp} F\] where \(r_{\perp}\) = perpendicular distance (lever arm)
• Newton's Second Law for rotation: \[\sum \tau = I\alpha\]
• Sign convention: Counterclockwise torque typically positive, clockwise negative

Rotational Dynamics and Energy

• Rotational kinetic energy: \[KE_{rot} = \frac{1}{2}I\omega^2\]
• Total kinetic energy (rolling object): \[KE_{total} = KE_{trans} + KE_{rot} = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2\]
• For rolling without slipping: \[v = r\omega\] \[KE_{total} = \frac{1}{2}mv^2 + \frac{1}{2}I\left(\frac{v}{r}\right)^2\]
• Work done by torque: \[W = \tau \Delta\theta\]
• Rotational power: \[P = \tau \omega\]
• Angular momentum: \[L = I\omega\] SI unit: kg·m²/s
• Conservation of angular momentum (when net external torque = 0): \[L_i = L_f\] \[I_i \omega_i = I_f \omega_f\]
• Torque and angular momentum: \[\sum \tau = \frac{dL}{dt}\]

Gravity and Orbital Motion

Universal Gravitation

• Newton's Law of Universal Gravitation: \[F_g = \frac{Gm_1m_2}{r^2}\] where \(G\) = 6.67 × 10^-11 N·m²/kg², \(m_1\) and \(m_2\) = masses, \(r\) = distance between centers
  Force is always attractive
• Gravitational field strength: \[g = \frac{GM}{r^2}\] where \(M\) = mass of planet/star, \(r\) = distance from center
• At Earth's surface: \[g = \frac{GM_E}{R_E^2} \approx 9.8 \text{ m/s}^2\] where \(R_E\) = Earth's radius
• Weight at distance r from Earth's center: \[W = \frac{GM_E m}{r^2}\]
• Gravitational potential energy (general form): \[PE_g = -\frac{Gm_1m_2}{r}\] Negative sign indicates attractive force; PE = 0 at r = ∞

Orbital Motion

• Orbital speed (circular orbit): \[v = \sqrt{\frac{GM}{r}}\] where \(M\) = mass of central body, \(r\) = orbital radius
• Centripetal force equals gravitational force: \[\frac{mv^2}{r} = \frac{GMm}{r^2}\]
• Orbital period: \[T = \frac{2\pi r}{v} = 2\pi\sqrt{\frac{r^3}{GM}}\]
• Kepler's Third Law: \[T^2 = \frac{4\pi^2}{GM}r^3\] For objects orbiting same body: \(\frac{T_1^2}{T_2^2} = \frac{r_1^3}{r_2^3}\)
• Escape velocity: Minimum speed to escape gravitational field \[v_{esc} = \sqrt{\frac{2GM}{r}}\] At Earth's surface: \(v_{esc} \approx 11.2\) km/s
• Total mechanical energy in orbit: \[E = KE + PE = \frac{1}{2}mv^2 - \frac{GMm}{r} = -\frac{GMm}{2r}\] Negative total energy indicates bound orbit

Simple Harmonic Motion

General SHM Equations

• Displacement in SHM: \[x(t) = A\cos(\omega t + \phi)\] or \[x(t) = A\sin(\omega t + \phi)\] where \(A\) = amplitude, \(\omega\) = angular frequency, \(\phi\) = phase constant
• Velocity in SHM: \[v(t) = -A\omega\sin(\omega t + \phi)\] or (if using sine for position): \[v(t) = A\omega\cos(\omega t + \phi)\]
• Maximum velocity: \[v_{max} = A\omega\] Occurs at equilibrium position (x = 0)
• Acceleration in SHM: \[a(t) = -A\omega^2\cos(\omega t + \phi) = -\omega^2 x\]
• Maximum acceleration: \[a_{max} = A\omega^2\] Occurs at maximum displacement (x = ±A)
• Restoring force: \[F = -kx = ma = -m\omega^2 x\] Proportional to displacement, opposite in direction
• Angular frequency: \[\omega = 2\pi f = \frac{2\pi}{T}\] SI unit: rad/s
• Period: \[T = \frac{2\pi}{\omega} = \frac{1}{f}\]
• Frequency: \[f = \frac{1}{T} = \frac{\omega}{2\pi}\]

Mass-Spring System

• Angular frequency: \[\omega = \sqrt{\frac{k}{m}}\] where \(k\) = spring constant, \(m\) = mass
• Period: \[T = 2\pi\sqrt{\frac{m}{k}}\]
• Frequency: \[f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}\]
• Total energy (conserved in ideal SHM): \[E = \frac{1}{2}kA^2 = \frac{1}{2}mv^2 + \frac{1}{2}kx^2\]
• Kinetic energy: \[KE = \frac{1}{2}mv^2 = \frac{1}{2}kA^2\sin^2(\omega t + \phi)\]
• Potential energy: \[PE = \frac{1}{2}kx^2 = \frac{1}{2}kA^2\cos^2(\omega t + \phi)\]
• Maximum potential energy: \(PE_{max} = \frac{1}{2}kA^2\) (at x = ±A)
• Maximum kinetic energy: \(KE_{max} = \frac{1}{2}kA^2\) (at x = 0)

Simple Pendulum

Valid for small angles (θ <>

• Period: \[T = 2\pi\sqrt{\frac{L}{g}}\] where \(L\) = length of pendulum, \(g\) = gravitational acceleration
  Independent of mass and amplitude (for small angles)
• Frequency: \[f = \frac{1}{2\pi}\sqrt{\frac{g}{L}}\]
• Angular frequency: \[\omega = \sqrt{\frac{g}{L}}\]
• Restoring force (small angles): \[F = -mg\sin\theta \approx -mg\theta = -mg\frac{x}{L}\] where \(x\) = horizontal displacement
• Maximum speed (at bottom): \[v_{max} = \sqrt{2gL(1 - \cos\theta_0)} \approx \sqrt{gL\theta_0^2}\] where \(\theta_0\) = maximum angular displacement

Physical Pendulum

• Period (extended object): \[T = 2\pi\sqrt{\frac{I}{mgd}}\] where \(I\) = moment of inertia about pivot, \(d\) = distance from pivot to center of mass

Fluids at Rest (Hydrostatics)

Pressure

• Pressure definition: \[P = \frac{F}{A}\] SI unit: Pascal (Pa) = N/m²
  Also: 1 atm = 101,325 Pa ≈ 1.01 × 10⁵ Pa
• Pressure vs. depth in fluid: \[P = P_0 + \rho gh\] where \(P_0\) = pressure at surface, \(\rho\) = fluid density (kg/m³), \(g\) = 9.8 m/s², \(h\) = depth
• Gauge pressure: \[P_{gauge} = P - P_{atm} = \rho gh\] Pressure above atmospheric
• Absolute pressure: \[P_{abs} = P_{atm} + P_{gauge}\]
• Pressure difference between two points: \[\Delta P = \rho g\Delta h\]

Density and Specific Gravity

• Density: \[\rho = \frac{m}{V}\] SI unit: kg/m³
  Water: \(\rho_{water} = 1000\) kg/m³ = 1 g/cm³
• Specific gravity (SG): \[SG = \frac{\rho_{substance}}{\rho_{water}}\] Dimensionless ratio

Pascal's Principle

• Statement: Pressure applied to enclosed fluid is transmitted undiminished to every point in the fluid
• Hydraulic lift: \[\frac{F_1}{A_1} = \frac{F_2}{A_2}\] \[F_2 = F_1 \frac{A_2}{A_1}\] where subscript 1 = input, subscript 2 = output
• Displacement relationship: \[A_1 d_1 = A_2 d_2\] where \(d\) = displacement

Buoyancy

• Archimedes' Principle: Buoyant force equals weight of displaced fluid \[F_b = \rho_{fluid} V_{displaced} g\] where \(V_{displaced}\) = volume of object submerged
• For floating object (equilibrium): \[F_b = W_{object}\] \[\rho_{fluid} V_{submerged} g = m_{object} g = \rho_{object} V_{object} g\]
• Fraction submerged: \[\frac{V_{submerged}}{V_{total}} = \frac{\rho_{object}}{\rho_{fluid}}\]
• Apparent weight in fluid: \[W_{apparent} = W_{actual} - F_b = mg - \rho_{fluid} V_{object} g\]
• For object denser than fluid (sinking): \(F_b <>
• For object less dense than fluid (floating): \(F_b = W\) (partially submerged)
• For object with same density as fluid: \(F_b = W\) (suspended at any depth)

Fluids in Motion (Fluid Dynamics)

Continuity Equation

Conservation of mass for incompressible fluid:

• Equation of continuity: \[A_1 v_1 = A_2 v_2\] or \[Av = \text{constant}\] where \(A\) = cross-sectional area, \(v\) = fluid velocity
• Volume flow rate (Q): \[Q = Av = \frac{V}{t}\] SI unit: m³/s
• Mass flow rate: \[\frac{dm}{dt} = \rho Av\]
• Interpretation: Fluid flows faster in narrower sections

Bernoulli's Equation

Conservation of energy for ideal fluid flow (incompressible, non-viscous, streamline flow):

• Bernoulli's equation: \[P_1 + \frac{1}{2}\rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g h_2\] or \[P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}\] where \(P\) = pressure, \(\rho\) = density, \(v\) = velocity, \(h\) = height
• Terms:
  \(P\) = pressure energy per unit volume
  \(\frac{1}{2}\rho v^2\) = kinetic energy per unit volume
  \(\rho gh\) = potential energy per unit volume

Special Cases of Bernoulli's Equation

Horizontal flow (h₁ = h₂):

• \[P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2\]
• Interpretation: Pressure decreases where velocity increases

Static fluid (v₁ = v₂ = 0):

• \[P_1 + \rho gh_1 = P_2 + \rho gh_2\]
• Reduces to: \(P = P_0 + \rho gh\)

Torricelli's theorem (efflux from opening):

• Speed of fluid exiting hole at depth h below surface: \[v = \sqrt{2gh}\] Same as free fall velocity from height h

Viscosity and Poiseuille's Law

• Viscous force: Resistance to flow in real fluids
• Poiseuille's Law (flow through cylindrical pipe): \[Q = \frac{\pi r^4 \Delta P}{8\eta L}\] where \(Q\) = volume flow rate, \(r\) = pipe radius, \(\Delta P\) = pressure difference, \(\eta\) = viscosity coefficient, \(L\) = pipe length
• Key relationships:
  Flow rate proportional to \(r^4\) (very sensitive to radius)
  Flow rate proportional to pressure difference
  Flow rate inversely proportional to viscosity and length

Reynolds Number

• Reynolds number (determines flow type): \[Re = \frac{\rho v D}{\eta}\] where \(D\) = characteristic length (e.g., diameter), \(\eta\) = viscosity
• Laminar flow: Re < 2000="" (smooth,="" orderly="">
• Turbulent flow: Re > 4000 (chaotic flow)
• Transitional flow: 2000 < re=""><>

Additional Important Relationships

Coefficient of Restitution

• Definition (for collisions): \[e = \frac{|v_{1f} - v_{2f}|}{|v_{1i} - v_{2i}|}\] Ratio of relative speeds after and before collision
• Elastic collision: e = 1
• Perfectly inelastic collision: e = 0
• Real collisions: 0 < e=""><>

Hooke's Law Applications

• Work to stretch/compress spring: \[W = \int_0^x F \, dx = \int_0^x kx \, dx = \frac{1}{2}kx^2\]
• Average force during compression/extension: \[F_{avg} = \frac{kx}{2}\]

Terminal Velocity

• Terminal velocity (free fall with air resistance): When drag force equals weight: \[F_{drag} = mg\] For spherical object: \(v_{terminal} \propto \sqrt{\frac{mg}{r^2}}\)
  At terminal velocity: \(a = 0\)

Surface Tension

• Surface tension (γ): Force per unit length at liquid surface
  SI unit: N/m
• Excess pressure in bubble: \[\Delta P = \frac{4\gamma}{r}\] where \(r\) = bubble radius
• Capillary rise: \[h = \frac{2\gamma\cos\theta}{\rho gr}\] where \(\theta\) = contact angle, \(r\) = capillary radius