UNIT 4 - Fields and Further Mechanics

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• Momentum = mv Force = Δ(mv)/Δt Impulse = Δ(mv)

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• Elastic - Collision where there is no loss of kinetic energy Totally Inelastic - Where the colliding objects stick together Partially Inelastic - There is a loss in kinetic energy

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• v=(2πr)/T Angular displacement = θ = 2πft Angular speed = ω = 2πf                                    = (2π)/T

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• a = v²/r = ω²r Centripetal Force = F = mv²/r                                                = mω²r

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• Amplitude - Maximum displacement from equilibrium Time Period - Time for one complete cycle of oscillations Frequency - Number of cycles per second Phase Difference = (2π∆t)/T

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• Simple Harmonic Motion is defined as oscillating motion in which acceleration is proportional and opposite to displacement a = -(2πf)²x x = Acos(2πft)

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• When: Adding Extra Mass - The frequency is reduced as it increases the inertia of the system. Using Weaker Springs - The frequency is reduced as the force at any given displacement will be less so acceleration and speed will be less.

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• E = 1/2(mv²)     = 1/2k(A²-x²) Hence: v = ±(2πf)√(A²-x²)

1. Damping

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   • Light Damping - Amplitude gradually decreases over many oscillations Critical Damping - Returns to equilibrium in the shortest time possible and then stops Heavy Damping - The displacement gradually decreases to equilibrium with no oscillation

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• When a periodic force is applied to an oscillating system, the system undergoes forced oscillations. When the periodic force occurs at the same frequency as the natural frequency of the system, maximum displacement is achieved

1. Resonance

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   • A system is in resonance when the applied frequency equals the natural frequency

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• g=f/m Radial Field - Where the field lines are like spokes on a wheel, always directed to the centre Uniform Field - Where the field strength is the same in magnitude and direction throughout the field

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• V=W/mPotential Gradient = ΔV/Δr The gravitational potential at a point is the work done per unit mass to move a small object from infinity to that point

1. Potential Gradient

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   • Equipotentials are lines of constant potential, like contours on a map The potential gradient at a point in a gravitational field is the change of potential per metre at that point

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• F=(G*m1*m2)/r²

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• For a point mass M: g=F/m=(GM)/r² For a spherical mass M of radius R: F = (GMm)/r² g=(GM)/r² Gravitational potential near a spherical planet: V=(-GM)/r

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• Speed of a planet: v=(GM)/r r³/T² = (GM)/4π²

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• E=F/Q Between two parallel plates: E=V/d

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• V=Ep/Q The electric field strength is equal to the negative of the potential gradient

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• F=1/(4πε0) * (Q1Q2)/r²

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• E=Q/(4πε0*r²)

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• C=Q/V

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• E = 1/2QV

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• ΔQ/Q = -Δt/CR Q = Q0e^(-t/RC) Time Constant = RC

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• The motor effect takes place when current cuts across a magnetic field, resulting in a force perpendicular to the plane that can be worked out with Fleming's Left Hand Rule. The force is greatest when the wire is at right angles to the magnetic field F=BIL

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• F=BQv

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• BQv = (mv²)/r r = (mv)/(BQ)

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• An induced emf can be increased by moving the wire faster in a field, by using a stronger magnet, or by making the wire into a coil with more turns. The direction of motion, field and current can be found using Fleming's Right Hand Rule

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• Coils - At the north end of a coil, current travels anticlockwise and at the south end vice versa Lenz's Law - The direction of the induced current is always such as to oppose the change that causes the current Faraday's Law - The induced emf in a circuit is equal to the rate of change of flux linkage through the circuit EMF = Blv

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• In an alternating current generator, the amount of emf being produced at any time is reliant on the angle between the coil of wire and the magnetic field ε = ε0 sin2πft

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• Vs/Vp = Ns/Np A step up transformer has more turns on the secondary coil than on the primary coil and voltage is therefore stepped up. The opposite is true of a step down transformer Efficiency = (IsVs)/(IpVp) * 100 Electrical Power remains equal