Oscillations

Question 1

Simple Harmonic Motion

Answer 1

• Oscillation caused by a force acting to repeatedly restore a moving object to its equilibrium position
• acceleration is always proportional to its displacement, and is directed at the equilibrium position (away from displacement)
• defining equation is F=-kx (F=force, x= displacement, k= constant)
• an object is at rest at the extremes of its oscillation, and at maximum velocity in the midpoint (equilibrium position)

Question 2

Equations of SHM

Answer 2

• Force =-k*displacement
• acceleration= -(angular velocity)^2*displacement
• angular velocity = 2pifrequency
• displacement = Amplitudecos(angular velocitytime)
• velocity = -amplitudeangular velocitysin(angular velocity*time)
• acceleration = -Amplitudeangularvelocity^2cos(angular velocitytime)
• Force = -mass(angular velocity)^2displacement
• Max Velocity = angular velocity* amplitude
• Max acceleration = angular velocity^2*amplitude
• angular velocity = sqrt(k/mass) = 2pi/Time period

Question 3

Springs

Answer 3

• springs oscillate with time period T=2pi*sqrt(mass/k)

- springs must be light and can be vertical or horizontal

Question 4

Pendulums

Answer 4

• a simple pendulum is a small, dense bob on a light inextensible spring
• horizontal component of a springs oscillation is F=mgsin(theta)
• If theta is small then sin(theta)=theta so F is proportional to displacement. This makes it SHM so T=2pi*sqrt(l/g)
• in perfect SHM a doesn’t depend on Amplitude so a= -g(theta)

Question 5

how to derive equations of shm

Answer 5

• x=Amplitudecos(angular velocitytime)
• derivative of x= v = -angular velocityAmplitudesin(angular velocity*time)
• at the ends of the oscillation (when displacement is maximum), velocity is 0. When displacement is at 0 velocity is maximum
• derivative of v = a = -angular velocity^2amplitudecos(angular velocitytime) = -angular velocity^2x

Question 6

Energy in shm

Answer 6

• kinetic energy of oscillator = 1/2massmax velocity = 1/2m(angular velocity*amplitude)
• when velocity is max potential energy is min, so total energy of system is kinetic energy (1/2mangular velocity^2*amplitude^2)
• as Ek decreases Ep increases by an equal amount so total energy of the system stays the same, assuming there is no loss of energy to surroundings or damping

Question 7

Types of oscillation

Answer 7

• Free oscillation = no external force acts on the system after initial ‘push’
• Damped oscillation = energy is lost to surroundings and away from the oscillation
• Forced oscillation = Force is applied continuously so that the frequency of the oscillation = frequency of the source
• a 100% free oscillation is impossible to achieve, however energy that is lost can be gradually restored. In a clock a spring is coiled up that releases energy back into the oscillation over time.

Question 8

Resonance

Answer 8

• natural frequency of a pendulum = 1/Time period
• as time period is dependent only on length of pendulum, when different lengths of pendulum are forced to oscillate by one driving pendulum, the pendulums of similar lengths will oscillate more
• this is because they will have a similar natural frequency, and as such absorb more energy from the driving oscillator. This is resonance, and when it is achieved it leads to very large amplitudes

Question 9

Damping and resonance

Answer 9

• When damped the resonant frequency (at which amplitude is highest) of an oscillator is lower than it’s natural frequency
• however the frequency at which the most energy is absorbed is still the natural frequency

Question 10

Uses of resonance

Answer 10

• Radios are tuned by altering the capacitance of the receiver circuit until it’s natural frequency matches with the frequency of the radio station you want. This means the most energy is absorbed by that oscillation and it is picked up as a strong signal
• Frequency of microwaves is close to the natural frequency of water, so they oscillate more and with more energy, heating food
• An MRI scanner aligns all protons in a body with a magnetic field then sends bursts of radiowaves. Different parts of the body have different natural frequencies and so will absorb different amounts of energy. When the energy is re-emitted the signal strength tells you which part of the body it came from

Question 11

Negative impacts of resonance

Answer 11

• causes bridges to have large sways so dampers must be fitted to dissipate energy
• car engines avoid resonance so the whole vehicle doesn’t shake
• ductile materials are used to build wind turbines so that they absorb energy in a hysteresis loop as they vibrate to stop large amplitudes building up

Question 12

SHM graphs

Answer 12

• displacement-time graphs for an oscillating object are cos waves, and the gradient at any point shows velocity at that point
• velocity-time graphs for an oscillating object are -sin waves, and the gradient at any point is acceleration at that point