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ROTATIONAL EQUILIBRIUM AND ROTATIONAL DYNAMICS ➔ The relation between radians and degree is 2π π...

ROTATIONAL EQUILIBRIUM AND ROTATIONAL DYNAMICS ➔ The relation between radians and degree is 2π π θᵣₐ𝑑ᵢₐ ₛ = 360° (θ𝑑ₑ𝑔ᵣₑₑₛ) = 180° (θ𝑑ₑ𝑔ᵣₑₑₛ) ➔ The relation between radians and degree is 1 𝑟𝑒𝑣 θᵣₑᵥₒₗᵤₜᵢₒ ₛ = 360° (θ𝑑ₑ𝑔ᵣₑₑₛ) ★ Angular velocity, ( ω ) - Rate of change of an angle, In symbols, this is: ★ Angular position/angular displacement, ( 𝜃 ) ∆θ - Describes the rotational position of the body. ∆ω = ∆𝑡 - When a body rotates, a line can be drawn from the axis of rotation to any point within the body. ★ Angular acceleration ( α ) - Change in angular velocity per unit of time: ∆ω ∆α = ∆𝑡 - The angle in radians through which a point or line has been rotated in a specified sense about specific axis  → ★ Rigid Body 𝐿 = angular momentum - A solid composed of a collection of particles that 𝐼 = inertia remain static relative to each other and relative to ω = angular velocity the axis of rotation. Regardless of the forces acting on a body, it maintains its original shape and size. ★ Torque → - The torque, τ is perpendicular to the plane ★ Newton’s First Law of Motion on Rotating Bodies → → containing the position, 𝑟 and the force, 𝐹 and has - States that unless hindered by an external influence, magnitude a rigid body rotating about a fixed axis will remain → → → rotating at the same rate within the axis | | | τ = 𝑟 𝑥 𝐹 = 𝑟𝐹𝑠𝑖𝑛θ | - The SI unit of torque is N⋅m. ★ Moment of Inertia - It is positive when it causes a counterclockwise - Quantity that impedes changes in an object’s rotation and is negative when it causes a clockwise rotational state of motion. rotation. 2 𝐼 = 𝑚𝑟 ★ Static Equilibrium m = mass - If the following conditions are to be satisfied, then a r = distance from the axis of rotation rigid body is said to be in equilibrium: 𝐼 = inertia Σ = 0, Στ = 0 ★ Angular Momentum - When the body is at rest, it is in static equilibrium. - The larger the moment of inertia is, the harder it is to change the state of motion of a rigid body by a external torque → 𝐿 = 𝐼ω GRAVITY AND SATELLITE MOTION ★ Acceleration due to gravity - The variable g is referred to as the acceleration of ★ Gravity gravity - Force by which a planet or other body draws objects - Its value is 9.8 m/s² on Earth toward its center. 𝐺𝑀ₑₐᵣₜₕ 𝑔= 2 - Earth's gravity is what keeps you on the ground and 𝑟 what makes things fall. 𝑔 = acceleration of gravity - The force of gravity keeps all of the planets in orbit 24 𝑀ₑₐᵣₜₕ = mass of earth / 5. 98 × 10 𝑘𝑔 around the sun. −11 2 2 G = gravitational constant / 6. 67 × 10 𝑁𝑚 /𝑘𝑔 r = distance of an object from the center of the ★ Newton’s Law of Universal Gravitation earth / 6371 km - States that “Every particle of matter in the universe attracts every other particle with a force that is ★ Gravitational Field directly proportional to the product of the masses of - Responsible for the force on a body. the particles and inversely proportional to the - Originate from all the massive bodies and result in square of the distance between them.” the attractive pull known as the gravitational force 𝐺𝑚₁𝑚₂ 𝐹𝑔 = 2 of the body. 𝑟 - The gravitational force per unit mass that would be Fg = magnitude of the gravitational force exerted on a small mass at that point. 𝑚₁, 𝑚₂ = mass of objects 1 and 2 𝐺𝑀 G = gravitational constant / 6. 67 × 10 −11 2 𝑁𝑚 /𝑘𝑔 2 𝑔= 2 𝑟 r = distance between the two masses ★ Gravitational Potential Energy ★ The Law of Periods/Harmonies - The gravitational potential energy of a pair of - The square of the orbital period of a planet is masses, m₁ and m₂, that are a distance r apart is proportional to the cube of the semi-major axis of −𝐺𝑚1𝑚2 its orbit 𝑈𝑔 = 𝑟 2 3 2 3 𝑇 ∝ 𝑎 𝑜𝑟 𝑇 ∝ 𝑟 ★ KEPLER’S THREE LAWS - The ratio of the squares of the periods (T²) of any ➔ In the early 1600s, Johannes Kepler proposed three two planets is equal to the ratio of the cubes of laws of planetary motion. their average distances from the sun (R³). ➔ The laws are the accurate description of the motion 2 3 2 2 𝑇1 𝑅1 𝑇1 𝑇2 of any planet and any satellite = ;= = 2 3 3 3 𝑇2 𝑅2 𝑅1 𝑅2 ★ The Law of Ellipses - The path of the planets about the sun is elliptical in - The average distance value is given in astronomical shape, with the center of the sun being located at units where 1 a.u. is equal to the distance from the one focus. earth to the sun: 11 1 𝑎𝑢 = 1. 4957 𝑥 10 𝑚 ★ The Law of Equal Areas - An imaginary line drawn from the center of the sun - The orbital period is given in units of earth-years to the center of the planet will sweep out equal where 1 earth year is the time required for the areas in equal intervals of time. earth to orbit the sun: 3. 156 𝑥 107 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 ★ Net Centripetal Force acting upon a satellite ★ Acceleration of Satellite - If the satellite moves in circular motion, then the - The acceleration value of a satellite is equal to the net centripetal force acting upon this orbiting acceleration of gravity of the satellite at whatever satellite is given by the relationship location that it is orbiting 2 𝐺𝑀𝑐𝑒𝑛𝑡𝑟𝑎𝑙 𝑀𝑠𝑎𝑡𝑣 𝐹𝑛𝑒𝑡 = 𝑎= 2 𝑅 𝑅 - This net centripetal force is the result of the ★ Orbital Period Equation gravitational force that attracts the satellite towards - The final equation that is useful in describing the the central body and can be represented as motion of satellites is Newton's form of Kepler's 𝐺𝑀𝑠𝑎𝑡𝑀𝑐𝑒𝑛𝑡𝑟𝑎𝑙 third law. 𝐹𝑔𝑟𝑎𝑣 = 2 2 2 𝑅 𝑇 4π 3 = 𝐺𝑀𝑐𝑒𝑛𝑡𝑟𝑎𝑙 𝑅 - Since 𝐹𝑔𝑟𝑎𝑣 = 𝐹𝑛𝑒𝑡 , 2 𝑀𝑠𝑎𝑡𝑣 𝐺𝑀𝑠𝑎𝑡𝑀𝑐𝑒𝑛𝑡𝑟𝑎𝑙 𝑅 = 2 𝑅 ★ Orbital Speed Equation 𝐺𝑀𝑐𝑒𝑛𝑡𝑟𝑎𝑙 𝑣= 𝑅 sPERIODIC MOTION AND SIMPLE HARMONIC MOTION expressed as seconds per cycle Frequency, f ★ Periodic motion - Number of cycles in a unit of time. - The word periodic simply means something that - The SI unit of frequency is the hertz, this unit is repeats at regular intervals. named in honor of the German physicist Heinrich - Any motion that repeats the same path – Hertz round-and-round, back-and-forth, or similar The angular frequency, ω ★ Oscillatory motion - Represents the rate of change of an angular - A type of periodic motion in which a particle or set quantity of particles moves back and forth. 2π - The basic principle of oscillation maintains that an ω = 2π𝑓 = 𝑇 oscillating particle returns to its initial state after a - The relationship between frequency and period is certain period of time. 1 1 𝑓= 𝑇 , 𝑇= 𝑓 TERMS IN PERIODIC MOTION Amplitude of the motion ( A ) ★ Simple harmonic motion (SHM) - Maximum value of the position of the particle in ➔ Oscillatory motion for a system in which the farther either the positive or negative x direction. the object moves from the center, the more it is pulled back. Period ( T ) - Time for one cycle. ➔ An object performing this must have a restoring - The SI unit is the second, but it is sometimes force upon it that seeks to return it to its  equilibrium position which is directly proportional ➔ The maximum values of the magnitudes of the to the object’s displacement velocity and acceleration are 𝑘 2 𝑘 𝑣𝑚𝑎𝑥 = ω𝐴 = 𝑚 𝐴; 𝑎𝑚𝑎𝑥 = ω 𝐴 = 𝑚 𝐴 ➔ It is given by Hooke’s Law 𝐹 =− 𝑘𝑥 ★ Damping ➔ Maximum displacement is the amplitude ➔ Restraining of vibratory motion, such as mechanical oscillations, noise, and alternating electric currents, ➔ The angular frequency ω, period T, and frequency f by dissipation of energy of a simple harmonic oscillator are given by 𝑘 2π 𝑚 1 𝑘 ➔ Underdamping - exhibits oscillatory motion with ω = 𝑚 , 𝑇 = ω = 2π 𝑘 , 𝑎𝑛𝑑 𝑓 = 2π 𝑚 decreasing amplitude m = mass of the system k = force constant ➔ Critical damping condition exhibits no oscillation and the motion returns to the equilibrium position ➔ Displacement as a function of time is given by after being displaced and released. 𝑥(𝑡) = 𝐴𝑐𝑜𝑠(ω𝑡 + ϕ) ➔ Overdamping condition will also not exhibit ➔ The velocity is given by oscillation but the motion returns to the equilibrium 𝑣(𝑡) =− 𝐴ω𝑠𝑖𝑛(ω𝑡 + ϕ) position more slowly after being displaced from this position and released. ➔ The acceleration is 𝑎(𝑡) =− 𝐴ω𝑐𝑜𝑠(ω𝑡 + ϕ) WAVES ★ Transverse waves - Waves in which the particles of the medium move ★ Wave perpendicular to the direction of the propagation of - Disturbance that propagates energy from one place the wave to another without transporting any matter. ★ Mechanical Waves - Waves that need a medium in order to propagate ★ Electromagnetic waves - Waves that do not need a medium in order to ★ Wave pulse propagate - Single isolated propagating disturbance - The wave speed ( v ), depends on the type of wave ★ Longitudinal wave and the properties of the medium - Waves in which the particles of the medium move parallel to the propagation of the wave ★ PARTS OF A WAVE ★ Periodic wave - Wave number is - The wavelength ( λ ) is the distance over which the 2π 𝑘= λ wave pattern repeats. - The amplitude ( A ) is the maximum displacement of a particle in the medium. - The product of λ and f equals the wave speed. ★ Sinusoidal wave - Special periodic wave in which each point moves in simple harmonic motion. ★ Wave function y ( x, t ) - Gives a mathematical description of a wave. For a wave on a string, y is the displacement of a particle at time t and position x. - The wave function for a sinusoidal wave moving in the positive x-direction can be written as 𝑦(𝑥, 𝑡) = 𝐴𝑐𝑜𝑠(𝑘𝑥 − ω𝑡) - The wave function for a sinusoidal wave moving in the negative x-direction can be written as 𝑦(𝑥, 𝑡) = 𝐴𝑐𝑜𝑠(𝑘𝑥 + ω𝑡) SOUND AND WAVES transferred by the wave through or onto the surface. ★ Sound - A longitudinal wave in a medium 𝑊 ➔ Its SI unit is 2 𝑚 ★ Audible waves ➔ A point source emits sound waves of power P - Lies within the range of sensitivity of the human ear. equally in all directions ( isotropically ) - The human ear is sensitive to waves in the frequency range from about 20 to 20,000 Hz, called 𝑃 the audible range ➔𝐼= 𝐴 P = time rate of energy transfer ( power ) ★ Infrasonic waves A = area of the surface intercepting the sound - Frequencies below the audible range. - Elephants can use infrasonic waves to communicate ➔ The intensity at a distance r from a point source that with each other, even when separated by many emits sound waves of power P equally in all kilometers. directions ( isotropically ) is 𝑃 𝐼= 2 ★ Ultrasonic waves 4π𝑟 2 - Frequencies above the audible range wherein, 4π𝑟 = area of the sphere - The ultrasonic sound it emits is easily heard by dogs, although humans cannot detect it at all ➔ The sound level in decibels (dB) is defined as 𝐼 β = (10 𝑑𝐵) 𝑙𝑜𝑔 𝐼0 ★ Intensity, ( I ) ➔ The average rate per unit area at which energy is  ★ Superposition of waves ★ Interference - The resultant displacement of two or more Occurs when two waves overlap while traveling overlapping waves is the algebraic sum of the along the same medium displacements of the individual waves. Constructive interference - when the displacements due to the two waves are in the same direction ★ A standing wave - A pulsating stationary pattern caused by the interference of harmonic waves of equal amplitude and wavelength traveling in opposite directions. ★ Nodes and Antinodes Destructive interference - when the displacements - If the displacement at a point in space remains zero are in opposite directions as a wave travels through, that point is a node - If the displacement at a point in space varies over the greatest range as a wave travels through, that point is an antinode ★ Doppler Effect Change in the observed frequency of a wave when the source or the detector moves relative to the transmitting medium For sound the observed frequency 𝑓𝑜is given in terms of the source frequency 𝑓𝑠 by 𝑣±𝑣𝑜 𝑓𝑜 = 𝑣±𝑣𝑠 𝑓𝑠 wherein, 𝑓𝑜 = frequency of the observer 𝑓𝑠 = frequency of the source 𝑣 = speed of sound in a medium 𝑣𝑠 = velocity of the source ➔ ( + ) if it moves away to observer ➔ ( + ) if it moves towards observer 𝑣𝑜 = velocity of the observer ➔ ( + ) if it moves towards source ➔ ( + ) if it moves away to source FLUID MECHANICS - For gases, the standard is the density of oxygen at 0°C and pressure of 1 atm: ★ Density 3 1. 43 𝑘𝑔/𝑚 ➔ Describes how much space an object or substance takes up (its volume) in relation to the amount of matter in that object or substance (its mass). ★ Pressure ➔ Force ( F ) per unit area ( A ) ➔ Mass per unit volume 𝐹 𝑃= 𝐴 𝑚 ρ= 𝑉 m = object’s mass ➔ The pressure P ( absolute pressure ) at depth h in a V = its volume liquid is given by: 𝑃 = 𝑃0 + ρ𝑔ℎ ★ Specific gravity here, - is the ratio of the density of a material to a standard 𝑃0 = pressure at the surface density. It is a unitless quantity. ρ𝑠𝑢𝑏𝑠𝑡𝑎𝑛𝑐𝑒 ρ = fluid’s density 𝑆𝐺𝑠𝑢𝑏𝑠𝑡𝑎𝑛𝑐𝑒 = ρ𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑔 = acceleration due to gravity ρ𝑔ℎ = gauge pressure - The standard density for liquids is the density of ➔ lf the surface is open to the air, the pressure is equal water at 4°C: to atmospheric pressure 𝑔 𝑘𝑔 1 3 𝑜𝑟 1000 3 5 𝑐𝑚 𝑚 𝑃0 = 𝑃𝑎𝑡𝑚 = 1 𝑎𝑡𝑚 = 1. 013 × 10 𝑃𝑎  ➔ lf the surface is covered by a gas, ★ Archimedes’ Principle 𝑃0 = 𝑃𝑔𝑎𝑠 ➔ When a body is completely or partially immersed in a fluid, the fluid exerts an upward force on the body equal to the weight of the fluid displaced by the ➔ For a closed surface, body. 𝐹 𝑃= 𝐴 ➔ Buoyant force - is the upward force exerted by a fluid. ➔ The pressure is the same at any two points at the same level in the fluid. ➔ 𝐹 =𝑤 =𝑚 𝑔 =ρ 𝑉 𝑔 𝐵 𝑓 𝑓 𝑓 𝑓 ➔ Pressure in a liquid at any location is exerted in where, equal amounts in all directions 𝐹𝐵 = buoyant force ρ𝑓 = density of the displaced fluid ➔ Pressure is not volume dependent. The shape of the 𝑉𝑓 = volume of the displaced fluid container does not matter. 𝑔 = acceleration due to gravity ★ Pascal’s Principle - Pressure applied to a fluid in a closed container is ★ Bernoulli’s Principle transmitted equally to every point of the fluid and ➔ As the speed of a moving fluid increases the to the walls of the container pressure within the fluid decreases 𝐹1 𝐹2 𝐴1 = 𝐴2 1 2 1 2 ➔ 𝑃1 + 2 ρ𝑣1 + ρ𝑔𝑦1 = 𝑃2 + 2 ρ𝑣2 + ρ𝑔𝑦2 TEMPERATURE AND HEAT TRANSFER ★ Fahrenheit temperature scale ★ Temperature - Named after Dutch physicist Daniel Gabriel - Measure of the hotness or coldness of an object Fahrenheit - An object’s temperature is measured using a - Based on the freezing point (32°F) and boiling point thermometer, which is dependent on the properties (212°F) of water at standard atmospheric pressure. and behavior of matter during a temperature change. ★ Kelvin temperature scale - Named after British physicist William Thomson ★ Zeroth Law of Thermodynamics - Based on the changes in pressure and consequently - Objects with the same temperature are in thermal temperature of gases in fixed volume containers. equilibrium. - By extrapolation from pressure — temperature - Thermometers are able to measure temperatures of graphs, the zero temperature (absolute zero or 0 K) other objects by achieving thermal equilibrium with is determined to be at –273.15°C. the object. - The Kelvin scale is not degree-based and is thus written without the degree sign TEMPERATURE SCALES ★ Conversion among three scales ★ Celsius temperature scale 9 𝑇𝐹 = 5 𝑇𝐶 + 32° - Named after Swedish astronomer Andres Celsius 5 - This is also known as the centigrade scale, because 𝑇𝐶 = 9 (𝑇𝐹 − 32°) it is based on the freezing point (0°C) and boiling point (100°C) of water and has 100 degrees in 𝑇𝐾 = 𝑇𝐶 + 273. 15 between. aTHERMAL EXPANSION ➔ Thermal insulators - materials that prevent the transfer of heat ★ Linear expansion ★ Conduction ➔ ∆𝐿 = α𝐿0∆𝑇 Transfer of heat by direct contact where, ∆𝐿 = change in length in m The amount of heat per unit time transferred from 1 α = coefficient of linear expansion in °𝐶 an object of high temperature to an object of lower 𝐿0 = original length in m temperature can be measured as heat current ( H ) ∆𝑇 = change in temperature in °C Heat current is expressed in Joule per second, or Watt ( W ) ★ Volume expansion ➔ ∆𝑉 = β𝑉0∆𝑇 𝑘𝐴∆𝑇 𝑘𝐴(𝑇𝐻−𝑇𝐶) 𝐻 = 𝐿 = 𝐿 where, 3 where, ∆𝑉 = change in volume in 𝑚 1 𝑘 = thermal conductivity β = coefficient of volume expansion in °𝐶 𝐴 = area 3 𝑉0 = original volume in 𝑚 𝑇𝐻 = temperature at the warm end ∆𝑇 = change in temperature in °C 𝑇𝐶 = temperature at the cold end 𝐿 = length / thickness HEAT TRANSFER ➔ Thermal conductors - materials that permit the transfer of heat ★ Convection Emissivity - defined as the ratio of the energy Transfer of heat by a fluid such as water/air emitted by a material to a perfect emitter; has a value between 0 and 1 and depends on the material 𝐻 = ℎ𝐴∆𝑇 where, h = convection coefficient 𝐴 = area ∆𝑇 = change in temperature ★ Radiation Transfer of heat by electromagnetic radiation 4 𝐻 = 𝐴𝑒σ𝑇 where, 𝐴 = area of the radiating body e = emissivity σ = stefan boltzmann constant ∆𝑇 = temperature The Stefan–Boltzmann constant is equivalent to −8 𝑊 5. 67 × 10 2 4 𝑚𝐾