MOTION NEWTON’S LAWS (TAKEN FROM CENTRE OF MASS) 1. Inertia ⇒ An object at rest will remain at rest ⇒ An object in motion will remain in uniform motion ⇒ Unless acted upon by a net unbalanced force 2. F = ma ⇒ Acceleration is directly proportional (and in line with) the net force acting on an object ⇒ Acceleration is indirectly proportional to mass 3. When object A exerts a force on object B, object B exerts an equal and opposite force on object A NET FORCE Calculated from addition of vectors: 1. 1D: Addition of magnitude 2. 2D: Vectors headtotail or resolution into two perpendicular components UNIFORM CIRCULAR MOTION ⇒ f = 1 T , v = 2π T or v = 2πrf ⇒ Velocity is tangential to the motion path ⇒ Magnitude of acceleration: ⇒ a = v 2 r or a = 4π 2 r T 2 or a = 4π 2 rf 2 ⇒ Centripetal acceleration: Acceleration is always toward the centre of the circle ⇒ Velocity and acceleration are NOT constant (always changing) ⇒ Velocity is perpendicular to acceleration ⇒ Net Force MUST be toward the centre of the circle (centripetal force) to sustain circular motion Banked Track ⇒ 3 forces: 1. Normal force 2. Weight force 3. Friction force ⇒ F cent = Σ F net = N + W F cent = mv 2 r = mg tanθ = v 2 r = g tanθ ⇒ Resolve Normal and Friction forces into vertical and horizontal components ⇒ Sum of vertical components = 0 ⇒ Sum of horizontal components = ma ⇒ Maximum speed: when friction reaches maximum ⇒ Design speed: when friction = 0, v = gr tanθ , where θ is the banking angle Vertical Circular Motion ⇒ Object mass m, tension in string, T: ⇒ Highest: T + mg = mv 2 r ⇒ Lowest: T − mg = mv 2 r PROJECTILE MOTION ⇒ 2D motion under a constant force (gravity, or weight) ⇒ Horizontal component of velocity vector remains constant ⇒ Vertical component of velocity vector is affected by gravity, constant acceleration of g downwards. ⇒ For horizontal component: ⇒ a = 0 , v = u = Vcosθ , s = ut ⇒ V = speed at angle θ to horiz. ⇒ For vertical component: ⇒ Use rules for rectilinear motion: v = u + at s = 1 2 (u + v ) s = ut + 1 2 at 2 s = vt − 1 2 at 2 v 2 = u 2 + 2as ⇒ u = Vsinθ , a = −g MOMENTUM & ENERGY Impulse ⇒ Impulse = change in momentum: ⇒ I = Δp , FΔt = mv − mu ⇒ Conservation of momentum: ⇒ Total momentum before = total momentum after ⇒ m 1 u 1 + m 2 u 2 = m 1 v 1 + m 2 v 2 ⇒ When one object gains momentum, the other loses momentum by the same amount. (The total remains constant) ⇒ Δp 2 = −Δp 1 , I 2 = −I 1 Work ⇒ Work is done by one system on another system during which the former exerts a force on the latter. (energy transfer) ⇒ Change in Kinetic Energy ⇒ Results from work done by net force on an object. ⇒ Fs = 1 2 mv 2 − 1 2 mu 2 ⇒ When an object moves in a gravitational field kinetic energy changes to gravitational energy, and vice versa. ⇒ Total energy remains constant ⇒ E k1 + U g1 = E k2 + U g2 Gravitational Potential Energy ⇒ @ the Earth’s surface: ⇒ U g = mgh ⇒ ΔU g = mgΔh ⇒ ΔU g is given by the area under a force distance, or fielddistance graph Spring Energy ⇒ Hooke’s Law: when an object interacts with a Hookean spring, kinetic energy is changed to elastic potential energy and vice versa. Total energy remains constant. ⇒ F = kx ⇒ E k1 + U e1 = E k2 + U e2 ⇒ U e = 1 2 kx 2 ⇒ Area under forceextension graph is change in elastic potential energy ⇒ ΔU e = 1 2 k( x 2 ) 2 − 1 2 k( x 1 ) 2 ⇒ k = YA l Elastic collision ⇒ Elastic collision: ⇒ Total kinetic energy before and after collision is equal. (Energy is conserved) ⇒ During collision some kinetic energy is converted to elastic potential energy, and then back again ⇒ Inelastic collision: ⇒ Energy after collision is less than energy before. (Energy is lost) ⇒ During collision, some kinetic energy is converted to heat and sound. Gravitational fields ⇒ Universal gravitational field: ⇒ g = GM r 2 ⇒ Gravitational force between and two objects: ⇒ F = GM 1 M 2 r 2 ⇒ Satellite Motion: ⇒ a = g ⇒ v 2 r = GM r 2 or 4 π 2 r T 2 = GM r 2 ⇒ ∴ v 2 r = GM or r 3 T 2 = GM 4π 2 ELECTRONICS GENERAL EQUATIONS ⇒ Power: ⇒ P = IV ⇒ P = V 2 R ⇒ P = I 2 R ⇒ V = IR ⇒ E = VIt = Pt ⇒ Q = It RESISTANCE ⇒ The ability of a conductor to resist the flow of electric current. Ohm’s Law ⇒ Ohm’s law states that for ohmic conductors, the resistance stays constant, when voltage and current vary. ⇒ V = IR , R = V I Resistance in Parallel ⇒ 1 R P = 1 R 1 + 1 R 2 + 1 R 3 + ... or R P = 1 1 R 1 + 1 R 2 + 1 R 3 + ... ⇒ R P = V AB I Resistance in Series ⇒ R S = R 1 + R 2 + R 3 + ... ⇒ R S = V AB I Nonohmic conductors ⇒ Diodes: ⇒ Device used to control current and voltage ⇒ Conducts when forward biased ⇒ Current drops to virtually 0 in reversebias ⇒ Thermistors: ⇒ Resistance varies with temperature ⇒ Transducers: ⇒ Change other forms of energy (heat, light, etc.) into electricity and vice versa. ⇒ Photonic Transducers: ⇒ Change light into electricity and vice versa. ⇒ Light Dependant Resistors (LDRs): ⇒ Resistance changes with the intensity of light it is exposed to ⇒ Photodiodes: ⇒ Conductivity changes with illuminating light intensity when in reversebias (photoconductive mode) ⇒ As light intensity increases, current (photocurrent) increases. ⇒ Forward biased mode is called photovoltaic mode. ⇒ Light Emitting Diodes (LEDs): ⇒ Emits light when forward biased. Light intensity increases with increasing forward current. POWER ⇒ P total = ΣP = P 1 + P 2 + P 3 + ... ⇒ P total = V AB I ⇒ P total = V AB 2 R total ⇒ P total = I 2 R total CURRENT ⇒ POTENTIAL DIFFERENCE/VOLTAGE ⇒ The change in electrical potential energy between two points. Voltage Dividers ⇒ A series connection of two or more resistors forms a voltage divider. The supply voltage ⇒ V 1 V 2 = R 1 R 2 ⇒ V out = R out R 1 + R 2 × V in Voltage Amplification ⇒ Voltage gain: ⇒ gain = ΔV out ΔV in ⇒ i.e.: gain is gradient of voltage inout graph. ⇒ Negative value for inverting, positive for noninverting. ⇒ If input signal exceeds maximum, clipping occurs. Clipping PHOTONICS Frequency Modulation ⇒ Modulation: ⇒ Changing the intensity of the carrier light wave to replicate the amplitude variation of the signal wave ⇒ Allows signals that are more robust and able to travel longer distances. ⇒ Demodulation: ⇒ Separation of a signal wave from the carrier wave. STRUCTURES AND MATERIALS FORCES Tension and Compression ⇒ When a When a structure/material is pulled at both ends/stretched, it is under tension. ⇒ When a structure/material is pushed at both ends/squashed, it is under compression. ⇒ Compression and tension forces are taken overall, i.e.: a material of nonuniform crosssectional area experiences uniform compression and tension. ⇒ Compression and tension can coexist in a structure. Shear ⇒ Where two opposing parallel forces in the same plane are applied to opposite sides of a structure/material, or, when two opposing rotational forces in the same plane are applied to a structure/material, it is experiencing a shear force. EFFECTS OF FORCES/ENERGY Stress, Strain and Young’s Modulus ⇒ Stress is experienced by any material subjected to a force. Because stress is inversely proportional to crosssectional area, thinner materials experience more stress (and ∴ more likely to fail) ⇒ σ = F A ⇒ σ ∝ 1 A , σ ∝ F 2.5 5 7.5 10 12.5 15 -2.5 2.5 carrier wave output wave signal wave -3 -2 -1 0 1 2 3 -3 -2 -1 1 2 3 General Diagrams Voltage Dividers Signal Modulation Voltage Amplification inout graph General Diagrams Projectile Motion General Diagrams