Transcript Newton*s second law of motion - crypt

10 CREATING MODELS
Simulating 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.
If an oscillator has this property it is a
Harmonic Oscillator.
Why is this so important in a
grandfather clock?!?
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?
B
C
A
3. What effect does doubling the mass
have on:
· the time for one oscillation?
· the frequency of the oscillations?
Looking carefully at oscillations
4. What forces act on the mass?
5. At what point are these forces
balanced?
B
C
A
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
Sinusoidal oscillation
+A
Phasor picture
s = A sin t
amplitude A
A
angle t
Page 16
0
time t
–A
periodic time T
phase changes by 2
f turns per 2 radian
second
per turn
 = 2f radian per second
Periodic time T, frequency f, angular frequency :
f = 1/T unit of frequency Hz
 = 2f
Equation of sinusoidal oscillation:
s = A sin 2ft
s = A sin t
Phase difference /2
s = A sin 2ft
s = 0 when t = 0
sand falling from a swinging pendulum leaves
a trace of its motion on a moving track
s = A cos 2ft
s = A when t = 0
t =0
A sinusoidal oscillation has an amplitude A, periodic time T, frequency f =
1
and a definate phase
T
Simple harmonic motion
• Derive velocity-time
and accelerationtime graphs from a
displacement-time
graph
• Explain the graphs in
terms of forces
acting
Analysing Oscillations
contrasting monochrome background
~1m
G-clamp
Using 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)
~1m
thin elastic cord
webcam sited to
cover at least positive
displacement
G-clamp
TASK
Plot graphs of
displacement,
velocity and
acceleration against
time.
Graph analysis...
TASK
From 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 accelerationtime traces
A body oscillates with SHM if its acceleration is
proportional to its displacement but oppositely
directed to it.
RECAP
How would that look graphically?
Deriving acceleration of a mass on a
spring
a=
𝑘
− 𝑠
𝑚
Can you work out where this
equation comes from?
a=
𝐹
𝑚
𝑎𝑛𝑑 𝐹 = −𝑘𝑠
Time period for a pendulum
Assuming a small angle θ
(less than ≈ 10⁰)
Restoring force F=-T(s/L)
(horizontal component of
tension)
θ
Length L
Restoring
force F
Displacement, s
Tension ≈ Weight (T=mg)
F=-mg(s/L)
a = - g(s/L)
T = 2𝜋
Tension T
Weight mg
𝑙
𝑔
A body oscillates with SHM if its acceleration is
proportional to its displacement but oppositely
directed to it.
Mathematically:
a𝛼 −𝑠
a = -constant x 𝑠
a=
𝑑𝑣
=-constant x 𝑠
𝑑𝑡
𝑑2𝑠
= -constant x 𝑠
2
𝑑𝑡
𝑑𝑣
𝑑2𝑠
= 2 = -constant x
𝑑𝑡
𝑑𝑡
𝑠
We already know all of this... What is the
constant?!?
Deriving equations for acceleration of
a mass on a spring. (assume SHM)
s
Equation for the displacement is:
S = Asin2Πft
Equations for acceleration of a mass
on a spring
a=
𝑘
− 𝑠
𝑚
also
Combining and rearranging
gives:
What would be equation
for T (time period)?
f=
1
2𝜋
T = 2𝜋
𝑘
𝑚
𝑚
𝑘
Summary (what you must know)
𝑎=
𝑑2𝑠
𝑑𝑡 2
2
= -(2π𝑓) 𝑠 =
T = 2𝜋
𝑚
𝑘
T = 2𝜋
𝑙
𝑔
𝑘
− 𝑠
𝑚
Mass on a spring
Pendulum
Mass on a spring
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
B
C
D
ma = kx
ma = -kx2
ma = -kx
ma = -k/x
mass m, acceleration a, displacement x and
constant k.
What is differential equation for SHM?
(comes from definition of SHM)
2
𝑑 𝑠
𝑑𝑡
=
2
𝑘
− 𝑠
𝑚
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 Euler
Swiss Mathematician/
Physicist
Explaining how it
works…
4 key steps….
1. Future displacements can be calculated from a knowledge
of displacement now and velocity now.
s = last s + vΔt = last s + Δs
2. Future velocities can be calculated from a knowledge of
velocity now and acceleration now.
v = last v + aΔt = last v + Δv
Where do we get acceleration (a) come from….
3. a = F/m (remember a =
4. F = -kxs
𝑑2 𝑥
)
2
𝑑𝑡
Dynamics of harmonic oscillator
How the graph starts
zero initial velocity would stay
zero if no force
velocity
force changes
velocity
How the graph continues
force of springs accelerates mass towards
centre, but less and less as the mass nears the
centre
change of velocity
decreases as
force decreases
new velocity
= initial velocity
+ change of
velocity
trace curves
inwards here
because of
inwards
change of
velocity
t
0
0
time
trace straight
here because no
change of
velocity
no force at centre:
no change of velocity
time
Constructing the graph
because of springs:
force F = –ks
t
change in displacement = v t
t
if no force, same velocity
and same change in
displacement
plus
extra change in
displacement from
change of velocity due
to force
extra displacement = –(k/m) s (t)2
acceleration = F/m
acceleration = –(k/m) s
change of velocity v
= acceleration  t
v = –(k/m) s t
extra displacement
= v t
Health warning! This simple (Euler) method has a flaw. It always changes
the displacement by too much at each step. This means that the oscillator
seems to gain energy!
SHM consolidation
Use 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 at
http://www.youtube.com/watch?v=ytBtbsEAXXc
Q1. 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