1. A-level Physics: Oscillations
Simple Harmonic Motion
Free and Forced Oscillations

2. Starter Activity
FLASHBACK: Explain what emission and absorption spectra are and how we can use them to determine the atmosphere of a star

3. Emission Spectra
When you heat gases to high temperatures, electrons move to higher energy levels
These then fall back to their ground state releasing a photon
If you split the light via diffraction you get a line emission spectrum
When white light passes through a cool gas an absorption spectrum results
Photons of the correct wavelength are absorbed by electrons to a higher energy level
These wavelengths correspond to elements that the atmosphere is made of

4. Question: when is displacement, at a maximum?
Basics
Simple harmonic motion is the repeated movement of a freely moving object either side of a midpoint. For instance, in the metronome shown to the right, the centre point is the midpoint and the bar fluctuates from either side. Other examples include a pendulum and a mass-spring system There is always a restoring force towards the centre. The size of which depends on the displacement and causes the pendulum to accelerate towards the centre
Question: when is displacement, at a maximum?
When at either side.

5. Basics
Conditions of SHM: “An object will exhibit SHM when the restoring force (and so acceleration) is directly proportional to the object’s maximum displacement/amplitude and in the opposite direction to this displacement”
Question: What would a graph of its motion look like?
A cosine wave
F= -kx
With F as the restoring force, k as a constand and x as displacement

6. Potential Energy to Kinetic Energy
TASK: Sketch a graph of energy against time for one whole cycle of a metronome. One line should be kinetic energy (Ek) and one potential energy (Ep)
Energy
Left
Midpoint
Right
Midpoint
Left

7. Characteristics of simple harmonic motion:
It is periodic oscillatory motion about a central equilibrium point (O)
the displacement is a sinusoidal function of time, it ranges from zero to a maximum displacement (amplitude, A)
Velocity is maximum when displacement is zero
Acceleration is always directed toward the equilibrium point, and is proportional to the displacement but in the opposite direction,
The period does not depend on the amplitude.
Velocity
Time O B C C
Positive amplitude C
Time
Displacement O
Equilibrium position O
Negative amplitude B B B

8. Period and Frequency T=1/f
The period (T) of an oscillator is the time taken for one whole oscillation and is related to frequency by:
T=1/f
Positive amplitude
Equilibrium position O
Negative amplitude

9. Questions 1
1.A body performs simple harmonic motion when its acceleration is proportional to its displacement. What additional condition must be satisfied for the oscillations of a pendulum to be simple harmonic?
The diagram shows a long pendulum which is oscillating between points A and B. The pendulum takes 5.72 s to swing from A to B. Calculate its period T.
T =
What is the frequency of the oscillation?
……………………………………………….
Force is opposite direction to the maximum displacement. A B

10. 2. A body oscillates with simple harmonic motion
2. A body oscillates with simple harmonic motion. On the axes below sketch a graph to show how the acceleration of the body varies with its displacement.
a/ ms-2
X / m

11. 2. A body oscillates with simple harmonic motion
2. A body oscillates with simple harmonic motion. On the axes below sketch a graph to show how the acceleration of the body varies with its displacement.
Remember, the acceleration is directly proportional to the displacement and is in the opposite direction!
a/ ms-2
X / m

12. At C, t=0 the displacement its maximum positive valueB A A C C
Displacement O
Time B C B B
At C, t=0 the displacement its maximum positive value
x = Amplitude (A)

13. The displacement is zero x = 0B A C C
Displacement O
Time B C B B
At O, t = T/4
The displacement is zero x = 0
θ = π / °

14. the displacement is maximum negative x = -AO B -A A C C
Displacement O
Time B C B B
At B, t = T/2
the displacement is maximum negative x = -A
θ = π °

15. the displacement is zero x = 0B A C C
Displacement O
Time B C B B
At O, t = 3T/4
the displacement is zero x = 0
θ = 3π / °

16. the displacement is maximum positive x = AB A C A C
Displacement O
Time B C B B
At C, t = T
the displacement is maximum positive x = A
θ = 2π °

17. θ At a random point M , the displacement is x since ω = θ /t  θ = ω tB x C C C
Displacement O A
Time θ B C B B x
At a random point M ,
the displacement is x
since ω = θ /t 
θ = ω t
θ = θ°
Using trigonometry
cos θ = x / A 
cos (ω t) = x / A
x = A cos (ω t)

18. C M O B x C C C
Displacement O
Time B C B B x
What is the displacement of a ball going through SHM between points A and B where the distance between them is 1 m and the angle is π/ 7
x = A cos (ω t) or
x = A cos (θ)
x = 0.5 cos (π/ 7) = m

19. As the displacement of a wave over time is a cosine wave, the velocity is a sine wave. When the displacement is zero, the velocity is at its highest
The velocity of the wave is given as:
v = -ω A sin (ω t)
The acceleration is due to the force which acts in the opposite direction, and as such is inversely directly proportional to displacement
a = -ω2 A cos (ω t)
Acceleration
Time A O B
since x = A cos (ω t)
a = -ω2 x

20. Starter Activity
FLASHBACK: Draw a ray diagram for a) a real and b) a virtual image using a converging lens.

21. Ray Diagrams

22. For a mass-spring system
F = - kx
where F is the restoring force (the force exerted by the spring)
k is the spring constant, the bigger the value of k, the stiffer the spring.
x is the extension in the spring due to the mass hanging from it.
The negative sign means that the extension is in the opposite direction to the force x F mg
This is related to angular velocity by: k
ω2 = m

23. k
ω2 = m k
ω = m
2 π
since ω = T
2 π k m k
2 π T =
ω = = m T
SO…
For a mass on a spring the bigger the mass, the bigger the period (slower oscillations)
the bigger the k value (the stiffer the spring) the smaller the period (faster oscillations)

24. the bigger the mass, the smaller the frequency (slower oscillations).k
ω2 = m k
ω = m
2 π f
since ω = k =
2 π f
ω = m 1 k f =
2 π m
the bigger the k value (the stiffer the spring), the higher the frequency (faster oscillations).
the bigger the mass, the smaller the frequency (slower oscillations).

25. Springs in series kseries = F/2x kseries = kx/2x kseries = k/2
Springs can be combined to carry a single load. The spring constant k will depend on how these springs are arranged. In here they are arranged in series.
The load is suspended from the lower spring but the force F acts on both springs. k k
kseries = F/2x
kseries = kx/2x
F applied 2x
kseries = k/2
two springs in series is equivalent to a spring that is half the stiffness

26. Springs in parallel kparallel = F/(F/2k) kparallel = 2k
In here they are arranged in parallel.
The force is now shared between the two springs, so each spring has a force of F/2.
the extension for each spring is F/2k k k
x/2
F applied
kparallel = F/(F/2k)
kparallel = 2k
two springs in parallel is equivalent to a spring that is twice the stiffness

27. Free and Forced Oscillations
If you stretch and release a mass on a spring it will oscillate at its natural frequency. If no energy is transferred to or from its surroundings (i.e. like dampening), it will keep oscillating at the same amplitude forever. Naturally this doesn’t happen in reality but we call this kind of oscillation a free oscillation

28. Free and Forced Oscillations
However, a system can be forced to oscillate by a periodic external force which has a frequency called the driving frequency as it is literally driving the movement! This is known as a forced oscillation

29. Resonance
As the driving frequency approaches the natural frequency, the system gains more energy and so oscillates with a rapidly increasing amplitude When the natural frequency=driving frequency then this is at its maximum and is said to be in resonance

30. Question: why might this have happened?
Uhoh…
Question: why might this have happened?

31. Question: What will the curve of the previous graph look like?
Damping
As objects oscillate in a real-life context, there will always be a loss of energy to the surroundings due to external forces such as friction. This also occurs in plastic materials as they absorb the energy Systems are often deliberately damped to stop them oscillating or to minimise resonance. The degree of damping can range from light damping where the amplitude slowly decreases or heavy damping
Question: What will the curve of the previous graph look like?

32. Damping Graphs

33. Over-damping and Critical Damping

34. Practice Questions
Complete the set of practice questions provided. Once complete, mark the questions and then annotate with the information that is missing. Subsequent to this, place any questions that you lost lots of marks for in your weakness flashcard book