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#### Oscillations - Simple Harmonic Motion

###### Describing Oscillations
• Displacement (x) is the instantaneous distance of the moving object from its mean position.
• Amplitude (A) is the maximum displacement from the mean position.
• Period (T) is the time taken for one complete oscillation.
• Frequency (f) is the number of oscillations per unit time.
• Angular Frequency (ω) is the rate of change of angular displacement.
• ω = 2πf
• Phase Difference (Φ) is the measure of how much one wave is out of step with another wave.
• Φ = 2π t/T
• where T is time period, and t is the time lag between waves.
###### Simple Harmonic Motion
• Simple Harmonic Motion is acceleration proportional to the displacement and directed towards a fixed point.
• Requirements for S.H.M.:
• A mass that oscillates
• A position where the mass is equilibrium
• A restoring force that acts to return mass to equilibrium; F ∝ -x
• Defining the equation of S.H.M.
• a = -ω2x • The negative sign in the equation shows that a and x are in opposite directions. a is always directed towards the mean position.
###### Equations of S.H.M.
• Displacement:
• x = xo sin ωt, OR x = xo cos ωt
• Either one of the above, depending on initial conditions
• Velocity:
• v = ±ω√(xo2 – x2)
• v = vo cos ωt, OR v = -vo sin ωt [xoω = vo]
• Maximum velocity is at the equilibrium position and the minimum, 0, is at extremes.
• Acceleration:
• a = -ω2x
• a = -ω2(xo sin ωt) OR a = -ω2(xo cos ωt)
###### Graphs of S.H.M ###### Energy in S.H.M • Kinetic Energy:
• v = ±ω√(xo2 – x2)
• Ek = 1/2 mv2
• 1/22(xo2 – x2)
• Total Energy:
• At x = 0, Ek is max and equal to total energy
• ∴ Etotal1/22(xo2 – 02)
• ∴ Etotal1/22xo2
• Potential Energy:
• Etotal = Ek + Ep
• ∴ Ep = Etotal – Ek
• ∴ Ep = 1/22xo2 – [1/22(xo2 – x2)]
• ∴ Ep 1/22x2
• Graphs: ###### Examples
1. The needle of a sewing machine oscillates vertically through a total distance of 22 mm. The oscillations are simple harmonic with a frequency of 4.5 Hz. The cloth being sewn is positioned 8.0 mm below the needle when it is at its maximum height. Calculate, for the point of the needle:

1. its maximum speed
• Maximum speed can be calculated using vo = xoω
• Firstly, we must find the angular velocity:
• ω =  2πf = 2 × π × 4.5 = 28.3 rad s-1
• Next, we must find the amplitude. As the total vertical distance is 22 mm:
• xo = 22 ÷ 2 = 11 mm
• Substitute the data into the first equation:
• vo = 28.3 × 11 × 10-2 = 0.311 ms-1
2. its speed as it moves downwards through the cloth
• To find the velocity at that point, use the equation: v = ω√(xo2 – x2)
• We need to find the displacement for when the needle passes through the cloth. • 11 – 8 = 3 mm
• Hence, substitute the values into the equation and calculate v:
• v = 28.3 × √(112 – 32) = 0.30 ms-1
###### Damping
• Damping is the loss of energy and a reduction in the amplitude of an oscillating system caused by a force acting in the opposite direction of the system’s motion (e.g. friction).
• Light Damping: The system oscillates about an equilibrium position with a decreasing amplitude over a period of time. • Critical Damping: The system does not oscillate, and it is the amount of damping required such that the system returns to its equilibrium position in the shortest possible time.
• Heavy Damping: The damping is so great that the displaced object never oscillates but returns to its equilibrium position very very slowly. Practical Examples of Damping ###### Natural Frequency and Resonance
• Natural Frequency, fo is the unforced frequency of oscillation of a freely oscillating object.
• Free Oscillation is oscillatory motion not subject to an external periodic driving force; it oscillates at its natural frequency. • Forced Oscillation is oscillation caused by an external driving force; its frequency is determined by the driving force.
• Resonance is the maximum amplitude of vibration when an impressed frequency equals the natural frequency of vibration.

Damping and Resonance

• Effects of damping on frequency response of a system undergoing forced oscillations:
• It decreases amplitude at all frequencies.
• It slightly decreases the resonant frequency.
• The resonant peak becomes flatter. Purposes of Resonance

• Examples of Useful Purposes of Resonance:
• Oscillation of a child’s swing.