Chapter 17 - Oscillations Flashcards
Features of oscillating objects
- Have an equilibrium position
- Are displaced by a force
- Speed up when moving towards the equilibrium position and slow down when moving away from it
- Will all eventually return to their equilibrium position
Phase difference (φ)
The difference in displacement of two oscillating bodies or the displacement of one oscillating body at different times
In terms of 2π
Angular frequency (ω)
2πf or 2π/t
Units rad s^-1, works largely the same way as angular velocity
Simple harmonic motion
A form of oscillation where the acceleration of the object is directly proportional to the displacement and is always in the opposite direction
SHM acceleration
a = -(ω^2 x)
SHM time period and amplitude
They are independent as increasing the amplitude increases the average speed of the swing so the period does not change
SHM displacement-time graph
If the displacement is a maximum at t=0, it will be a cosine graph
If the displacement is 0 at t = 0, it will be a sine graph
SHM acceleration-time graph
An inverted displacement-time graph
SHM displacement formulae
x=Acos(ωt)
x=Asin(ωt)
Use cos when the displacement at t = 0 is not 0
SHM velocity formula
v = +/- ω x √(A^2 - x^2)
SHM maximum velocity formula
vmax = ωA
Energy-displacement graph for an oscillating object in SHM
Two parabolic curves for Ek and Ep
At x = 0 the Ek is a maximum and Ep 0
At x = +/- A, the Ek is 0 and Ep a maximum
The total energy is constant
Total energy of a spring in SHM
1
– k A^2
2
Kinetic Energy of a Spring in SHM
1
– k (A^2 - x^2)
2
Damping
External forces acting on an oscillator opposing its velocity which reduce the amplitude of the oscillation.
Damping
External forces acting on an oscillator opposing its velocity which reduce the amplitude of the oscillation.
Light damping
Damping forces are small, amplitude reduces gradually, period of the oscillation almost unchanged
e.g. pendulum in air
Heavy damping
Damping forces are large, amplitude decreases significantly, period of oscillation increases slightly
e.g. pendulum in water
Very heavy damping
Damping forces are very large, no oscillating motion, oscillator just moves to equilibrium position
e.g. pendulum in treacle
Displacement-time graph light damping
Follows the same cosine shape with a gradually reducing amplitude
Displacement-time graph heavy damping
Displacement reduces to a slight negative before returning to the equilibrium position and remaining there
Displacement-time graph very heavy damping
Gradually decreasing amplitude before it continues along the equilibrium position (never becomes negative) - almost like a reciprocal curve
Free oscillation
Object displaced from equilibrium position and allowed to oscillate freely without external forces
Natural frequency of an oscillator
The frequency of the free oscillation
Forced oscillation
A periodic driver force is applied to an oscillator. The object will vibrate at the frequency of the driving force (driving frequency)
Driving
If a force has the same driving frequency as the natural frequency of the oscillator, it will cause the amplitude of the oscillation to greatly increase.
Barton’s pendulum
The oscillating bob will cause the amplitude of the cone pendulum with the same natural frequency as the bob’s driving frequency to increase
Resonance
The amplitude of an oscillation increases considerably where the driving frequency of a forced oscillation is the same as the natural frequency of an oscillating object
Resonance with no damping
The amplitude will keep increasing until the object fails
Resonance uses:
Clocks using the resonance of a pendulum
Musical instruments resonate to make music louder
Tuning circuits to select the correct frequency
Stronger damping effect on resonance
- Decreases the maximum amplitude of resonance
- Decreases the frequency at which resonance occurs
- The peak of the amplitude against frequency to widen
How does MRI work?
- The scanner has a strong magnetic field which causes hydrogen nuclei inside the body to precess
- Radio waves emitted from coils inside the scanner cause the nuclei to resonate and absorb energy
- When the radio waves are switched off, the hydrogen nuclei re-emit the energy gained as photons
- These are detected by receiving coils and the signals are processed by computers to create a 3D image
