Oscillations

Oscillations

• examples of of free oscillations:
• pendulum
• mass on spring
• an object oscillates when it oscillates back and forth repeatedly
• on either side of equilibrium position
• (if stopped it returns to equilibrium position)
• daily use:
• child swing
• vibrations of guitar string
• movement of pendulum
• free and forced:
• free - vibrates at particular frequency (natural frequency)
• forced - forced to vibrate e.g. engine
• amplitude, frequency and period
• phase difference
• to find, measure the difference between the two waves then divide it by the period to get the fraction, then multiply it by 2pi or 360
• simple harmonic motion
• requirements for s.h.m
• if a simple pendulum is not disturbed it will stay in its equilibrium position. When it is swinging, it must be pulled to one side of equilibrium position, the forces on the mass will be unbalanced.
• three requirements for smh for mechanical system:
• mass oscillates
• a position where mass is in equilibrium
• a restoring force that acts to return the mass to equilibrium position. F force to displacement of mass and is directed towards equilibrium point.

Diapositiva 2

• 
• The changes of velocity in s.h.m.
• As pendulum swings back and forth, its velocity is constantly changing. It has maximum speed as it travels through the equilibrium position.
• Graphical representations of s.h.m.
• A trolley tethered between two springs. - The motion sensor allows you to record how the displacement varies with time.
• At the equilibrium position (0 displacement) velocity is maximum but the acceleration is directly proportional to the displacement but is directed in the opposite direction because the (restoring) force is directed towards the equilibrium position.
• Displacement against time
• Mathematically this graph is a sine curve, d against t allows us to calculate the period T and amplitude A of the oscillations.
• Velocity against time graph
• the velocity v of the oscillator can be determined by finding the gradient of the displacement against time graph: v=(change in) x/(change in) t

Diapositiva 3

• Acceleration against time graph
• acceleration can be determined from the gradient of the velocity against time graph:
• a= (change in) velocity/(change in) time
• at the start the mass is in its equilibrium position, there's no restoring force so acceleration is zero. As it moves to the right, the restoring force acts towards the left, giving it a negative acceleration. The acceleration has it greatest value when the mass is displaced furthest away from the equilibrium position. (the graph is upside down compared to the displacement against t graph).
• acceleration ∝ -displacement
• a ∝ -x
• Frequency and angular frequency
• f=1/T
• there must be 2πf radians in unit time - this is the angular frequency.
• w (angular frequency)
• w=2πf or w=2π/T
• Equations of s.h.m.
• x=Asin(2πft)
• where A is the amplitude and f is the frequency
• sometimes the same motion is represented with a cos function.
• x=Acos(2πft)
• difference is the sin graph starts at x=0 and cos starts when x=A
• (2πft) is in radians
• Acceleration and displacement
• a=-(2πft)^2 *x
• a is directly proportional to x
• note the following acceleration against displacement graph:
• the graph is a straight line through origin
• negative slope
• gradient = (2πft)
• the gradient is independent of the amplitude of the motion. The frequency or period is independent of the amplitude

Diapositiva 4

• Maximum speed of an oscillator
• If an object is executing simple harmonic motion is has max speed when it passes the equilibrium position.
• vmax=(2πft)A
• vmax ∝ A
• vmax ∝ f
• increased frequency means a shorter period so it has to cover greater distance in shorter time, so it has a greater speed.
• Energy changes in s.h.m.
• There is a constant interchange of energy between two forms; potential and kinetic. In mass spring system, the springs store elastic potential energy. When it moves to equilibrium position it has max speed so it has max kinetic energy. The total energy in the system remains constant.
• Energy graphs
• Potential energy is maximum when displacement is maximum. Kinetic energy is maximum is maximum when displacement is zero. The total energy remains constant throughout.
• this graph shows that:
• kinetic energy is maximum when displacement is 0
• potential energy is maximum when displacement is max
• at any point PE + KE is the same (total energy is constant)
• Damped oscillations
• In principle oscillations can go on forever. In practice, they don't they die out gradually or rapidly. This happens because of friction.
• We describe these oscillations as damped. Their amplitude decrease according to a pattern. Exponential envelope decrease.
• The amplitude of the oscillations decays exponentially with time.- note that the frequency doesn't change as the amplitude decreases.
• Investigating damping
• A spring metal strip is clamped to a bench. A mass is attached to the end of the metal strip, this will oscillate if you displace it it one side. You can measure the decrease in the amplitude of the oscillations with a ruler.
• Amplitude against the time
• This graph shows exponential decrease, finding the half life by finding the time for half the amplitude.

Energy and damping - and resonance

• Damping can be very useful if we want to get rid of vibrations. For example, a car has springs making the ride more comfortable, it does this by having springs damped by shock absorbers.
• Damping is achieved by introducing friction in the mechanical system. In undamped oscillation the total energy remains constant.
• Resonance
• E.g. the tacoma narrows bridge.
• The swing with a child, the swing of the child has a natural frequency of oscillation.
• Observing resonance.
• If the forcing frequence matches the natural frequency, the amplitude of the oscillations can build up to get very big
• Barton's pendulums
• Several pendulums of different lengths hang from a horizontal string, each has a natural frequency. The driver pendulum at the end oscillates and causes one of the other pendulums to oscillate with a large amplitude as they have the same length of the string.
• Defining resonance
• When the forcing frequency matches the natural frequency the amplitude of the oscillations grows dramatically.
• In resonance energy is transferred from driver to resonating system, for example in the case of the Tacome Narrows bridge energy was transferred from the wind to the bridge.
• In resonance:
• natural frequency =driving frequency
• amplitude is maximum
• it absorbs the greates possible energy from the driver

Resonance

• 
• Damping is useful if we want to reduce the damaging effects of resonance. As the damping increases, the amplitude of resonance decreases. Resonance peak also becomes broader.
• Using Resonance
• Resonance can be a problem but it can be useful.
• Also, microwave cooking. The microwaves used have a frequency that matches the natural frequency of vibration of water molecules. The water molecules in the food vibrate with a greater amplitude cooking the food.
• MRI - magnetic resonance imaging