#### Oscillations - Simple Harmonic Motion

__Describing Oscillations__

__Describing Oscillations__

is the instantaneous distance of the moving object from its mean position.**Displacement (x)**is the maximum displacement from the mean position.**Amplitude (A)**is the time taken for one complete oscillation.**Period (T)**is the number of oscillations per unit time.**Frequency (f)**is the rate of change of angular displacement.**Angular Frequency (ω)**- ω = 2πf

is the measure of how much one wave is out of step with another wave.**Phase Difference (Φ)**- Φ = 2π
^{t}/_{T} - where T is time period, and t is the time lag between waves.

- Φ = 2π

__Simple Harmonic Motion__

__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 = -ω
^{2}x

- a = -ω
- 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.__

__Equations of S.H.M.__

- Displacement:
- x = x
_{o}sin ωt,**OR**x = x_{o}cos ωt - Either one of the above, depending on initial conditions

- x = x
- Velocity:
- v = ±ω√(x
_{o}^{2}– x^{2}) - v = v
_{o}cos ωt,**OR**v = -v_{o}sin ωt [x_{o}ω = v_{o}] - Maximum velocity is at the equilibrium position and the minimum, 0, is at extremes.

- v = ±ω√(x
- Acceleration:
- a = -ω
^{2}x - a = -ω
^{2}(x_{o}sin ωt)**OR**a = -ω^{2}(x_{o}cos ωt)

- a = -ω

__Graphs of S.H.M__

__Graphs of S.H.M__

__Energy in S.H.M__

__Energy in S.H.M__

- Kinetic Energy:
- v = ±ω√(x
_{o}^{2}– x^{2}) - E
_{k}=^{1}/_{2}mv^{2} - ∴
^{1}/_{2}mω^{2}(x_{o}^{2}– x^{2})

- v = ±ω√(x
- Total Energy:
- At x = 0, E
_{k }is max and equal to total energy - ∴ E
_{total}=^{1}/_{2}mω^{2}(x_{o}^{2}– 0^{2}) - ∴ E
_{total}=^{1}/_{2}mω^{2}x_{o}^{2}

- At x = 0, E
- Potential Energy:
- E
_{total }= E_{k }+ E_{p} - ∴ E
_{p }= E_{total }– E_{k} - ∴ E
_{p }=^{1}/_{2}mω^{2}x_{o}^{2 }– [^{1}/_{2}mω^{2}(x_{o}^{2}– x^{2})] - ∴ E
_{p }=^{1}/_{2}mω^{2}x^{2}

- E
- Graphs:

__Examples__

__Examples__

- 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:- its maximum speed
- Maximum speed can be calculated using v
_{o }= x_{o}ω - Firstly, we must find the angular velocity:
- ω = 2πf = 2 × π × 4.5 = 28.3 rad s
^{-1}

- ω = 2πf = 2 × π × 4.5 = 28.3 rad s
- Next, we must find the amplitude. As the total vertical distance is 22 mm:
- x
_{o }= 22 ÷ 2 = 11 mm

- x
- Substitute the data into the first equation:
- v
_{o }= 28.3 × 11 × 10^{-2}= 0.311 ms^{-1}

- v

- Maximum speed can be calculated using v
- its speed as it moves downwards through the cloth
- To find the velocity at that point, use the equation: v = ω√(x
_{o}^{2}– x^{2}) - 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 × √(11
^{2}– 3^{2}) = 0.30 ms^{-1}

- v = 28.3 × √(11

- To find the velocity at that point, use the equation: v = ω√(x

- its maximum speed

__Damping__

__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.

: 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.**Critical Damping**: The damping is so great that the displaced object never oscillates but returns to its equilibrium position very very slowly.**Heavy Damping**

**Practical Examples of Damping**

__Natural Frequency and Resonance__

__Natural Frequency and Resonance__

is the unforced frequency of oscillation of a freely oscillating object.**Natural Frequency, f**_{o}is oscillatory motion not subject to an external periodic driving force; it oscillates at its natural frequency.**Free Oscillation**

is oscillation caused by an external driving force; its frequency is determined by the driving force.**Forced Oscillation**is the maximum amplitude of vibration when an impressed frequency equals the natural frequency of vibration.**Resonance**

**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.
- Tuning of a radio receiver – the natural frequency of a radio is adjusted so that it responds resonantly to a specific broadcast frequency.
- Using a microwave to cook food – It produces microwaves of frequencies equal to the natural frequency of water, causing the water molecules in the food to vibrate, thereby generating heat.
- Magnetic Resonance Imaging (MRI) is used in hospitals to create images of human organs.

- Examples of the Destructive Nature of Resonance:
- High-pitches sound waves can shatter fragile objects, e.g. shattering a glass when a soprano hits a high note.
- Buildings that vibrate at natural frequencies close to the frequency of seismic waves collapse during earthquakes.
- A car suspension system vibrates when going over bumps, which wold give large amplitude vibrations.