FC312E
PHYSICS
Theme: 2 Hour: 2 Waves and Optics
Module Learning Outcome
Academic Literacy
Determine change in energy in
SHM
Use theoretical concepts to analyse real-world
practices.
Explain phenomenon of resonance
Develop ability to read, comprehend and express
mathematical arguments
Apply in real-world contexts
Use subject-specific vocabulary effectively.
SIMPLE HARMONIC
MOTION
Outline (Hour 2)
• Energy is SHM
• Energy conservation
• Damped oscillations
• Resonance
• Forced oscillations
Energy in SHM
Energy in SHM
• By observing the motion of a simple pendulum we can
deduce that:
– The bob has its maximum gravitational potential energy
(GPE) at maximum displacement.
• At this point the velocity is zero for an instant. Thus
the kinetic energy (KE) is zero.
– The bob has its maximum
velocity as it passes through
the centre of oscillation.
• Thus its KE is maximum
and its GPE is a minimum.
GPEmax
GPEmax
KEmax
Energy in SHM
• In any oscillation there is a constant interchange between
kinetic and potential energy.
• The total energy remains constant provided no energy is
lost from the system.
• The graph shows how KE and GPE vary during one
oscillation.
𝑬𝒕𝒐𝒕𝒂𝒍 = 𝑬𝒌𝒊𝒏𝒆𝒕𝒊𝒄 + 𝑬𝒑𝒐𝒕𝒆𝒏𝒕𝒊𝒂𝒍
Energy in SHM
The total energy is constant provided there is no damping.
Speed in SHM equation from
Energy Conservation
If we compare two points of an oscillating spring’s motion.
First we note that the formula for potential energy in a spring
depends on displace x through equation:
1 2
𝐸𝑃 = 𝑘𝑥
2
So thinking about the maximum displacement point A where
the total energy ET=EP we get:
1 2
𝐸𝑇 = 𝑘𝐴
2
Speed in SHM equation from
Energy Conservation
• So if that is the total energy we can equate this to the sum of
the kinetic and potential energies at some displacement x:
1 2 1
1 2
2
𝑘𝐴 = 𝑚𝑣 + 𝑘𝑥
2
2
2
• Rearranging this equation for the velocity v at some
displacement x we get the important equation:
𝑘 2
2
𝑣 = (𝐴 − 𝑥 2 )
𝑚
𝑘
𝑣=±
𝐴2 − 𝑥 2 = ±𝜔 𝐴2 − 𝑥 2
𝑚
(Recall that:
𝜔=
𝑘
𝑚
for a spring)
(This equation now also applies for a pendulum)
Example
• Question: A 0.33kg pendulum bob is attached to a string 1.2m
long. What is the:
a)
Change in gravitational potential energy of the system as
the bob swings from point A to point B in figure below:
Hint: you will need to use
some basic trigonometry
to find the distance Δ𝑦
b)
Use this to find the velocity of the pendulum bob as it moves
through point B
Solution
a)
To get the change in gravitational potential Δ𝑈 energy
we need to find the change in height Δ𝑦 using
trigonometry.
If you look at the pendulum
picture the trigonometry tells
you:
Then we apply the usual
potential energy equation 𝑈 =
𝑚𝑔ℎ
∆𝑦 = 𝐿 − 𝐿𝑐𝑜𝑠𝜃 = 𝐿(1 − 𝑐𝑜𝑠𝜃)
∆𝑈 = 𝑚𝑔∆𝑦 = 𝑚𝑔𝐿 1 − 𝑐𝑜𝑠𝜃
= (0.33 kg)(9.81 m s-1)(1.2 m)(1 - cos35°)
∆𝑈 = 0.70 𝐽
b) Then we use energy conservation to
equate the potential energy (magnitude) at
A to the kinetic energy at B:
1
𝑚𝑣 2 = 0.70 𝐽
2
𝑣=
2 × 0.7
= 2.06 m𝑠 −1
0.33
Energy Assumptions
• Typically, a physicist will state that energy is always
conserved in a ‘closed system’.
– A ‘closed system’ means that no energy can be lost to the
surroundings.
• Alternatively, a physicist may state that energy is lost to the
surroundings (but total energy is always conserved).
– In this case, we say that the system is losing energy, but the
surroundings are gaining energy.
• This process is often called damping.
Energy in SHM
• A ‘real’ pendulum bob, or other oscillating system will lose
energy to its surroundings.
• This is called damping.
• Sources of this energy loss, or damping, include:
– Air resistance (or more generally, resistance to motion due to the
medium in which the body is oscillating); or
– Friction at the point of oscillation (e.g. the arm of a pendulum);
• Damping results in the amplitude of the oscillations dying
away over a period of time.
Damping
• As a real pendulum or spring oscillates it will
inevitably lose energy and the amplitude will
decrease.
• We any oscillating motion that loses enery
damped.
• There are three types of damping you should
know:
– Light damping
– Strong damping
– Critical damping
Damping
• Light damping occurs when the time period of the oscillation is
unchanged as damping takes place.
• Critical Damping is when the damping is just right to make the
oscillation return to equilibrium as fast as possible without any further
oscillations taking place, e.g. car suspension
•
Strong damping is
when the damping
stops all oscillation and
makes the system
return to equilibrium
very slowly.
Lightly damped
Forced Oscillations
• If something is oscillating without there being any external
forces acting on the system then it is called a free oscillation.
• Free oscillators always oscillate at a special frequency called
the natural frequency.
• What happens however if we try to create an oscillation by
applying a periodic force to an oscillating system?
‣
For example consider
a spring-mass
system acting under
an applied vibration:
Forced Oscillations
https://www.youtube.com/watch?v=nFzu6CNtqec
Resonance
• We find from forced oscillation experiments that the
amplitude of the forced oscillation increases and becomes
enormous when the applied frequency of vibration gets
close to the natural frequency of the oscillator.
• This maximum amplitude applied frequency is also called
the resonant frequency.
So the key result is that
we observe a
resonance when:
𝑓𝑎𝑝𝑝𝑙𝑖𝑒𝑑 ≈ 𝑓𝑛𝑎𝑡𝑢𝑟𝑎𝑙
Resonance
Resonance is a fascinating phenomenon we encounter all around us
and it can have profound effects. For example:
– Bridges can collapse (click on image to see video)
– Glasses can shatter when sound hits their natural frequency
(click on image to see video)
– Car components rattling at certain velocities
– Acoustic resonances in music
– In lasers: creation of coherent light by optical resonance in a
laser cavity
Resonance
https://www.youtube.com/watch?v=yFsgu3ClqHo