Chapter 17 - Oscillations and SHM

Question 1

define oscillatory motion

Answer 1

“oscillating motion occurs where an object starts in an equilibrium position and a force is applied, causing it to oscillate either side of its equilibrium position”

Question 2

explain the ‘phases’ of oscillatory motion

Answer 2

• an object is displaced from its equilibrium position
• it accelerates towards the equilibrium position and decelerates once it’s passed it
• it reaches its maximum displacement the other side of the equilibrium position and the process repeats

Question 3

define displacement and amplitude, give their symbols and units

Answer 3

Displacement - x/m - Distance with respect to direction from its equilibrium position

Amplitude - A/m - Maximum displacement from equilibrium position

Question 4

define period, frequency and phase difference, give symbols and units

Answer 4

Period - T/s - time taken to complete one full oscillation
Frequency - f/Hz - number of complete oscillations per unit time
Phase difference - Phi/ radians - the difference in phase of the cycles of the oscillations

Question 5

define angular frequency

Answer 5

“angular frequency describes the motion of an oscillating object through the rate of change of angle”

angular frequency = 2(pi)/period

Question 6

what are the equations for angular frequency

Answer 6

Omega = 2(pi)/t
Omega = 2(pi)f

Question 7

give the equation for SHM and identify the main points

Answer 7

a = (-)(omega^2) x

where omega^2 is constant:
- acceleration is directly proportional to displacement
- it acts in the opposite direction (towards the equilibrium position)
- maximum acceleration occurs at maximum displacement (amplitude)

Question 8

give the three main features of SHM

Answer 8

• an acceleration-displacement graph has a constant gradient pf Omega^2
• frequency and period are constant
• period of a pendulum in SHM is independent of amplitude

Question 9

describe the practical used to determine the frequency and period of an object in SHM

Answer 9

• set up a pendulum on a clamp stand or a spring with masses attached to a clamp
• pick a point as a fiducial marker
• set the pendulum swinging or spring oscillating and time n oscillations using the fiducial marker to help
• divide the time by n to calculate the period of an oscillation
• repeat for different amplitudes
• calculate an average

Question 10

what shape are the graphs for SHM (displacement against time)

Answer 10

• sinusoidal
• if there are no energy losses Amplitude is constant

Question 11

what are the two possible shapes of a displacement-time graph for SHM

Answer 11

• if starting from max displacement then cosine shape
• if starting from equil position then sine shape

Question 12

assuming we start from max displacement what is the shape of the velocity time graph for SHM

Answer 12

• the derivative of cosine = -sine
• so a -ve sine graph

Question 13

assuming we start from max displacement, what is the shape of the acceleration time graph for SHM, what is important about this

Answer 13

• the double derivative of Cosine = -cosine
• so a is directly proportional to -x

Question 14

what are the equations linking displacement and time for SHM

Answer 14

X = Acos(omega*t)
- if starting from max displacement
- if ans is -ve then it is on the other side of the equil position

X = Asin(omega*t)
- if from equil position
- assign one side +ve and one side -ve

Question 15

what is the equation for SHM linking velocity and displacement, amplitude and omega, what is the Vmax equation

Answer 15

V = +- omega Sqrt(A^2 - X^2)

Vmax = omegaA
- Because max velocity occurs at equil position (X=0)

Question 16

define damping

Answer 16

“an oscillation is damped when an external force that acts on the oscillator has the effect of reducing the amplitude of its oscillations”

Question 17

what are the key points about light damping, give an example

Answer 17

light damping:
- low energy losses
- period almost unchanged
- amplitude gradually decreases

e.g. air

Question 18

what are the key points of heavy damping and give an example

Answer 18

heavy damping:
- higher energy losses
- period increases slightly
- amplitude decreases significantly

e.g. water

Question 19

what are the key points of very heavy damping and give an example

Answer 19

Very heavy damping:
- no oscillatory motion
- slowly returns to equil position

e.g. syrup

Question 20

what generally happens to the kinetic energy of the oscillator in damping

Answer 20

KE —> other forms (usually heat)

Question 21

what do the graphs of the types of damping look like

Answer 21

• light damping = sinusoidal but decreasing amplitude (exponentially)
• heavy damping dips below the x axis comes slightly above then back to rest
• very heavy damping looks like a -ve exponential

Question 22

what is a free oscillation

Answer 22

“a free oscillation is where a mechanical system is displaced from its equilibrium position and oscillates without external forces”

frequency = natural frequency

Question 23

what is a forced oscillation

Answer 23

” a forced oscillation is one in which a periodic driver force is applied to an oscillator”

freq = freq of driver

Question 24

what is resonance

Answer 24

“Resonance occurs where the driving frequency is equal to the natural frequency of the object, this causes amplitude to increase dramatically”
“Amplitude at this point is at a MAXIMUM”

Question 25

describe Barton's pendulum experiment

Answer 25

- a wire is fixed across two points - D is a heavier brass bob pendulum - there are a number of different lengths paper cone pendulums - D is set swinging, acting as a driving force - if a pendulum has the same length as D it will resonate and have a much greater amplitude than the other pendulums

Question 26

what can we say about the energy in a system of an object moving with SHM

Answer 26

- total energy remains constant assuming no energy losses

Question 27

describe the energy transfers that occur in an oscillation of a pendulum

Answer 27

- at the amplitude the object has no KE, only potential energy, in this case GPE - as the pendulum falls it loses GPE but gains KE - as it moves through the equilibrium position, it has maximum KE and no GPE

Question 28

what are the differences in the sources of potential energy between a pendulum and a mass-spring system

Answer 28

if the mass-spring system is vertically: - potential energy is GPE and Elastic Potential if the mass-spring system is horizontally - potential energy is only Elastic Potential

Question 29

what does the graph of energy against displacement look like for an object in SHM

Answer 29

- two parabolas - one -ve, one +ve - the +ve shaped parabola is Ep such that at max displacement, Ep is max - the -ve shaped parabola is KE such that at max displacement, KE = 0 - total energy remains constant as a line running across the top

Question 30

explain a simple demonstration of energy transfer in SHM

Answer 30

- spring and glider on an air track - compress spring - Elastic potential = 1/2 k X^2 - this is transferred to the kinetic energy of the glider (1/2 mv^2)

Question 31

how can the air track demonstration explain the shape of the graph

Answer 31

at any point KE = total energy - Ep total energy = Ep max Ep max = 1/2 k A^2 so KE = 1/2 k A^2 - 1/2 k X^2 KE = 1/2 k (A^2-X^2) so both KE and EP have parabola shapes

Question 32

what do the Ep and KE against time graphs look like against time for a pendulum

Answer 32

if pendulum starts from 0 displacement: - Ep-time graph is like sin but only positive so Mod(sin) - KE-time graph is like V^2 so cos^2, in other words a cos graph shifted up

Question 33

give some examples of where resonance is used

Answer 33

- clocks use the resonance of a pendulum or quartz crystal to keep time - musical instruments have resonant casings and the air column inside instruments resonates - tuning circuits use resonance e.g. car radios - MRI scanners

Question 34

what does the graph of Amplitude of oscillation against driving frequency look like, give its key features

Answer 34

- a number of lines which rise up to a peak then drop away but not a smooth parabola - the lines represent different amounts of damping - higher damping gives a lower line with a broader peak at a lower frequency - the peak of the highest line corresponds to F0, the natural frequency of the object

Question 35

describe the effect of damping on resonance

Answer 35

As the amount of damping increases: - Amplitude of vibration at any frequency decreases - frequency of Max amplitude is lower - peak becomes flatter and broader

Question 36

describe how an MRI scanner works

Answer 36

- surrounding the scanner there are superconducting electromagnets and coils to produce radio waves - These provide a very strong magnetic field - hydrogen nuclei behave like tiny magnets and precess (sort of spin thing) - when radio waves interact with them they resonate - once the nuclei stop resonating they release the energy as different wavelength radio wave photons that are detected

Question 37

Define SHM

Answer 37

an oscillatory motion where the acceleration of the oscillator is directly proportional to its displacement and is directed towards a fixed / equilibrium point. + draw a graph

Question 38

what does the defining SHM graph look like

Answer 38

a graph of a = -w^2x so effectively y = -x but with gradient -w^2

Question 39

what to remember to put when talking about the effect of amplitude changes on SHM

Answer 39

period is independent of amplitude

Question 40

in what direction does the resultant force act on an object in SHM and why

Answer 40

towards the equilibrium position: reason one: - there are two forces acting, tension and weight, these cancel to make the horizontal force reason two: - the only acceleration occurs towards the equilibrium position therefore that is the direction of the resultant force

Question 41

what can we derive from a v^2 against x^2 graph

Answer 41

v^2 = w^2A^2 - w^2x^2 so y-int = (vmax)^2 grad = (angular frequency)^2

Question 42

what does a kinetic energy against time graph look like for SHM (if starting from max displacement)

Answer 42

like a sin^2(x) graph because if v = -sin(x) then Ek = f((-sin^2(x)) it looks like a cos graph shifted up

Question 43

what is the only thing that affects the period of a pendulum in SHM

Answer 43

the length of the pendulum

Question 44

is the net force acting on a mass in a mass-spring system ever equal to its weight

Answer 44

ONLY when it passes through the equil point

Question 45

what is the major difficulty in measuring angles of swing and period in an investigation and how can we solve it

Answer 45

- the angle of swing decreases with time due to the damping effect of air resistance - to solve this you can measure a fraction of a swing (e.g. 1/4 of a swing) ACCURATELY using a DATA LOGGER and scale up to find period

Question 46

where can resonance be a nuisance

Answer 46

- washing machines can have casings that resonate and cause loud vibrations - wind/walkers can cause bridges to resonate

Question 47

How to calculate the period of a simple pendulum when given length

Answer 47

T = 2(pi) sqrt(l/g)