Simple harmonic motion

conditions for simple harmonic motion

• oscillations

• same time period for each complete oscillation - independent to amplitude (for small angles)

• restoring force responsible for the motion is always directed towards the equilibrium position

• restoring force is directly proportional to diplacement

• the acceleration is directly proportional to the displacement and in the opposite direction to displacement

acceleration of an object oscillating in simple harmonic motion

$$a=-\omega^2x$$

acceleration is at a maximum when displacement is at a maximum

acceleration at a minimum at equilibrium

SHM displacement equation for oscillations from equailibrium

$$x=A\sin\left(\omega t\right)$$

SHM displacement equation for oscillation from amplitude

$$x=A\cos\left(\omega t\right)$$

Free Oscillations

• freely oscillating object oscillates with a constant amplitude because there is no friction acting on it

• only forces acting on it is the restoring force

• with friction, amplitude of oscillations would gradually decrease over many cycles to 0

shm displacement, velocity and acceleration graphs

• velocity time graph is pi/2 out of phase with displacement graph and acceleration graph

• displacement graph is pi rad out of phase with acceleration graph

Energy of an object in shm

• energy of the system changes from kinetic energy to potential energy and back again every half cycle after passing through equilibrium (by restoring force)

energy diplacement graph

• potential energy curve is parabolic Ep = ½ kx²

• kinetic energy curve is an inverted parabola Ek = Et-Ep = ½k(A²-x²)

• ETotal = Ek + Ep = 1/2 kA² which is the potential energy at the maximum displacement and the kinetic energy at 0 displacement

• two curves add together to give a horizontal line for the total energy

• 50% kinetic energy occurs at x = sqrt(0.5)A
  
  

damped oscillations

• oscillations in which dissipating forces cause the amplitude to decrease - transfer energy to the surroundings as thermal energy

light damping

• time period is independent of the amplitude

• each cycle takes the same length of time as the oscillations die away

• amplitude of oscillations gradually reduces by the same fraction each cycle

critical damping

• just enough to stop the system oscillating after it has been displaced from equilibrium and released

• system returns to equilibrium in the shortest possible time

heavy damping

• so strong that the displaced object returns to equilibrium much more slowly than if critically damped - no oscillating motion occurs

natural frequency

• frequency at which system oscillates without a periodic force being applied

forced vibrations

• periodic force applied to oscillating system

effect of changing frequency of periodic force on amplitude of oscillations

• amplitude depends on the frequency of the applied force

• as the applied frequency increases, the amplitude of the oscillations increases until it reaches a maximum amplitude at a particular frequency

• then the amplitude decreases again

phase difference between displacement and periodic force as frequency increases

• increases from 0 to $\frac{\pi}{2}$ rad at maximum amplitude

• increases from $\frac{\pi}{2}$ rad to $\pi$ rad as frequency increases further

Resonance

• system oscillating at max amplitude - limited only by damping - energy supplied by the periodic force is lost at the same rate due to damping when at a constant max amplitude

• at resonance, energy is transferred most efficiently from driver to oscillating system - maximum kinetic energy transferred into it

• periodic force is $\frac{\pi}{2}$ rad out of phase with displacement

• periodic force is in phase with velocity

• frequency at which this occurs is resonant frequency

• at equilibrium (max velocity) P=fv so most power transferred out of system by damping - most power can also be transferred into system by driving force on- most efficient

effect of damping on resonant frequency

• the heavier the damping, the smaller the maximum amplitude becomes at resonance (reduces amplitude of all frequencies) due to energy losses out of the system

• resonance peak broadens (max amplitudes more significantly reduced around peak)

• the further (and smaller) the resonant frequency is compared to the natural frequency of the system - damping slows the oscillations, increasing time period and decreasing frequency - resonant frequency must also lower to be in phase with velocity

• for an oscillating system with little or no damping, at resonance, the applied frequency of the periodic force being = the natural frequency of the system (resonant frquency)

• when the damping is not light, resonance occurs at a slightly lower frequency than the natural frequency

oscillating surface with object on it

• object remains in contact with surface when the vibrating surface accelerates down with an acceleration less than g

• above a specific frequency, the downwards acceleration of the surface becomes greater than g, meaning that the surface will move back down with an acceleration grater acceleration from max amplitude than the object can reach

• so object no longer in contact with surface as it moves down

two oscillators of different time periods in sm. time at which they will be in phase and number of full oscillations of each until they are in phase

• A: T=1.98

• B: T=2

• 1.98 × 100 = 198 so in phase after 100 oscillations of A

• 2 × 99 = 198 so in phase after 99 oscillations of B

• in phase at 198 seconds