10 CREATING MODELSSimulating reality• Learn how models are used

in physics to provide a simplified description of reality

• Set up and explore some physical models

Starter: What do you need if you are going to predict what the rate of growth in the economy is going to be in a year’s time?

Radioactive decay

• Recap principles: random nature and fixed decay probability

• Model decay processes, including sequential ones

Maths of radioactive decay

• Explain the term exponential process

• Derive and use the equations for radioactive decay

Starter:Examine the decay graphs on the board.What can you say about how the half lives of nuclide A and nuclide B compare? Explain your answer.

Gotham City car crime

There are 500 000 cars in Gotham City.In 2018, 12 000 are stolen and never recovered.

Q1. What is the annual car theft rate?Q2. What is the probability that a car will be stolen in any given year?Q3. After many years of crime, the number of cars has fallen to 400 000. What will be the annual theft rate now? What assumptions are you making?Q4. Can you devise an equation that relates the annual theft rate, the annual theft probability and the number of cars?Q5. “Holy cow, Batman! If there are only 400 000 cars left and they are being stolen at a rate of 12 000 per year, there will be none left in 33 years!” Why does Batman send Robin back to school?

What is the half life of iodine 131?How could the half life of uranium 238, which is 4.5 billion years, be measured?

Gotham City car crime

There are 500 000 cars in Gotham City.In 2018, 12 000 are stolen and never recovered.

Q1. What is the annual car theft rate?Q2. What is the probability that a car will be stolen in any given year?Q3. After many years of crime, the number of cars has fallen to 400 000. What will be the annual theft rate now? What assumptions are you making?Q4. Can you devise an equation that relates the annual theft rate, the annual theft probability and the number of cars?Q5. “Holy cow, Batman! If there are only 400 000 cars left and they are being stolen at a rate of 12 000 per year, there will be none left in 33 years!” Why does Batman send Robin back to school?

Half life of beer froths

• Investigate if decay of foams is an exponential process

Capacitor charge and discharge• Learn about storage of

charge by capacitors

• Learn and use the equation C = Q/V

• Derive equations for capacitor charge and discharge

• Know significance of the time constant (R x C)

Capacitor discharge

• Determine the time constant for capacitor discharging through a resistor

Modelling capacitor discharge

• Develop and use a simple software model for capacitor discharge

Energy storage in capacitors

• Learn how to determine the energy stored in a capacitor, and how to estimate the discharge power

Oscillations

• Investigate oscillations and explain the meanings of the terms isochronous oscillator and simple harmonic motion

Starter: How many examples of oscillating systems can you think of?

Isochronous?!?Isochronous (from Greek iso, equal + chronos, time):

It literally means regularly, or at equal time intervals.If the swing of a pendulum is isochronous, each swing

takes the same length of time whatever the amplitude of the swing.

Why is this so important in a grandfather clock?!?

If an oscillator has this property it is a Harmonic Oscillator.

B

A

C

Looking carefully at oscillations

1. When and where is the mass moving fastest / slowest / upwards / downwards?

2. When and where is the mass accelerating upwards / downwards / at a maximum value / zero?

3. What effect does doubling the mass have on:· the time for one oscillation?· the frequency of the oscillations?

B

A

C

Looking carefully at oscillations

4. What forces act on the mass?

5. At what point are these forces balanced?

6. When is the tension in the spring greatest? In which direction does the unbalanced force on the mass act at this point?

7. As the tension decreases, what will happen to the unbalanced force on the mass?

Language to describe oscillations

+A

0

–Aperiodic time Tphase changes by 2

Aangle t

Sinusoidal oscillation

time t

amplitude A

f turns persecond

= 2f radian per second

2 radianper turn

Phasor picture

s = A sin t

Periodic time T, frequency f, angular frequency :f = 1/T unit of frequency Hz = 2f

Equation of sinusoidal oscillation:s = A sin 2ft s = A sin t

Phase difference /2

s = A s in 2fts = 0 wh en t = 0

s = A c o s 2fts = A w hen t = 0

t = 0sand falling from a swinging pendulum leavesa trace of its motion on a moving track

A sinusoidal oscillation has an amplitude A, periodic time T, frequency f = and a definate phase1T

Page 16

Simple harmonic motion

• Derive velocity-time and acceleration-time graphs from a displacement-time graph

• Explain the graphs in terms of forces acting

contrasting monochrome background

thin elastic cordG-clamp G-clamp

~ 1 m ~ 1 m

webcam sited tocover at least positivedisplacement

Analysing OscillationsUsing a setup such as this.Oscillations have been recorded using a camera then analysed to record the displacement of the trolley against time. (click to open file)

TASKPlot graphs of displacement, velocity and acceleration against time.

Graph analysis... TASKFrom your graphs determine the following:--Period-Frequency-Maximum Amplitude-Maximum Acceleration-Maximum Velocity

Note down all numerical values and keep somewhere safe as we can compare these with theoretical values we will calculate in later lesson.

Maths of SHM

• Derive equations for simple harmonic oscillators such as the pendulum and the mass on spring

• Use equations in problem solving

Starter: The board sketch shows the velocity-time trace for an oscillator. Sketch it and add the corresponding displacement-time and acceleration-time traces

How would that look graphically?

A body oscillates with SHM if its acceleration is proportional to its displacement but oppositely

directed to it.

RECAP

Deriving acceleration of a mass on a spring

a =

Can you work out where this equation comes from?

a =

Time period for a pendulum

Length L

Tension T

θ

Weight mgDisplacement, s

Restoring force F

Assuming a small angle θ (less than ≈ 10⁰)

Restoring force F=-T(s/L)(horizontal component of tension)

Tension ≈ Weight (T=mg)

F=-mg(s/L) a = - g(s/L)

T =

A body oscillates with SHM if its acceleration is proportional to its displacement but oppositely

directed to it.

= -constant

Mathematically: a

a = -constant =-constant

= -constant

We already know all of this... What is the constant?!?

s

Deriving equations for acceleration of a mass on a spring. (assume SHM)

S = Asin2ΠftEquation for the displacement is:

Equations for acceleration of a mass on a spring

a =  also

Combining and rearranging gives:

f =

What would be equation for T (time period)?

T =

Summary (what you must know)

= -(

T =

T =

Mass on a spring

Mass on a spring

Pendulum

Modelling Simple Harmonic Motion• Develop and use a simple software model of SHM

Starter: Which of these differential equations describes a system that will oscillate with simple harmonic motion?

A ma = kxB ma = -kx2

C ma = -kxD ma = -k/x

mass m, acceleration a, displacement x and constant k.

What is differential equation for SHM? (comes from definition of SHM)

=

You can use this differential equation to estimate the displacement of an Oscillator (undergoing SHM) after a

given time using Euler’s algorithm (an iterative method).

Leonhard EulerSwiss Mathematician/

Physicist

Explaining how it works…

1. Future displacements can be calculated from a knowledge of displacement now and velocity now.

s = last s + vΔt = last s + Δs2. Future velocities can be calculated from a knowledge of velocity now and acceleration now.v = last v + aΔt = last v + ΔvWhere do we get acceleration (a) come from….3. a = F/m (remember a = 4. F = -kx

4 key steps….

s

Dynamics of harmonic oscillator

How the graph starts

force changesvelocity

0

time

t

zero initialvelocity

velocity would stayzero if no force

How the graph continues

force of springs accelerates mass towardscentre, but less and less as the mass nears thecentre

time

0

trace straighthere because nochange ofvelocity

no force at centre:no change of velocity

trace curvesinwards herebecause ofinwardschange ofvelocity

change of velocitydecreases asforce decreasesnew velocity

= initial velocity+ change ofvelocity

Constructing the graph

if no force, same velocityand same change indisplacementplusextra change indisplacement fromchange of velocity dueto force

= –(k/m) s (t)2

change in displacement = v t

t

t

extra displacement= v t

change of velocity v= acceleration tv = –(k/m) s t

because of springs:force F = –ks

acceleration = F/macceleration = –(k/m) s

Health warning! This simple (Euler) method has a flaw. It always changesthe displacement by too much at each step. This means that the oscillatorseems to gain energy!

extra displacement

SHM consolidationUse the white boards.Q1. Sketch a displacement-time trace for an oscillator where the

displacement = 0 when t=0.Q2. Add the corresponding velocity-time trace.Q3. Add the corresponding acceleration-time trace.Q4. Suppose the oscillator has a period of 0.1 seconds and an

amplitude of 0.05 m. Write the equation that describes how the displacement changes with time for this oscillator (Hint: using the format s = A sin 2πft or s = A cos 2πft)

Q5. Calculate the maximum acceleration and maximum velocity for the oscillator in Q4. Show your working.

Q6. In a certain oscillating system, the displacement is 0.55 m when the velocity is -0.32 ms-1. Calculate the displacement 0.05 seconds later. Assume that the period is much larger than 0.05 s.

Energy flow in oscillators

• Your task over the next 2 lessons is to produce an A3 spread on Energy in Oscillators

• Follow the task briefing sheet to do this.

Starter: Watch the video of the child on the swing athttp://www.youtube.com/watch?v=ytBtbsEAXXcQ1. At what frequency should you push to give large amplitude swings?Q2. Can you sketch a graph of how KE and PE change during the motion of the swing?

Resonance

• Explain the meanings of the terms natural frequency, driving frequency, forced oscillation and resonance

• Model the effects of damping on a resonance curve