3. Simple Harmonic Motion

4. Simple Harmonic Motion
Periodic Motion
Periodic motion is any motion that repeats itself.
Examples of periodic motion include:
Rise and fall of tides
Heart beat
Pendulum
Horizontal & vertical springs
Simple Harmonic Motion
For an object to be doing SHM the following two (three) conditions must apply to the periodic motion:
Acceleration is proportional to displacement
Acceleration directed towards the equilibrium position
There must exist a restorative force

5. acceleration is  to negative displacement
For an object undergoing SHM:
The velocity is NOT constant and will vary between Vmax and zero.
Since V is always changing there must be acceleration occurring
The acceleration is NOT constant and will vary between amax and zero.
x=-A
x=0
x=A
Vmax
a =0
V = 0
amax
V =0
amax
acceleration is  to negative displacement

8. V = 0, a = max
V = max, a = 0
V = 0, a = max

9. a a a a
Simple Pendulum
Equil. position

10. Essential features for Horizontal & Vertical Springs
x = 0 F m
A = xmax F
Equilibrium
Position x m
No difference in SHM calculations
between horizontal & vertical springs mg
(k) is the spring constant
Displacement (x) is measured from the equilibrium point
Amplitude (A) is the maximum displacement
A cycle is a full up-and-down or left-to-right motion
Period (T) is the time required to complete one cycle
Frequency (f) is the number of cycles completed per second
Similar features for a pendulum…

11. Periodic and Simple Harmonic Motion (SHM)
For these events discuss:
if they are periodic; and
if they approximate SHM.
Piston
Baby swing
Ski jumping
Heart beat
Bungy jumping

12. Periodic and approximates SHM
Periodic and Simple Harmonic Motion (SHM)
For these events discuss:
if they are periodic; and
if they approximate SHM.
Baby swing
Periodic and approximates SHM
The movement repeats over and over again in regular cycles
There is a restoring force which acts towards the equilibrium position (resting position)
The greater the displacement the greater the restoring force
The displacement and the restoring force are continually in opposite directions
Acceleration is a - displacement
Baby swing

13. Periodic and Simple Harmonic Motion (SHM) For these events discuss:
if they are periodic; and
if they approximate SHM.
Bungy jumping
In free fall the motion is not periodic and hence cannot be SHM because
After maximum displacement of the bungy cord the motion is periodic and approximates SHM because:
the movement repeats over and over again in regular cycles even though the amplitude decreases;
there is a restoring force which acts towards the equilibrium position. (final resting position);
the greater the displacement the greater the restoring force;
there are no unbalanced forces acting in the equilibrium position;
the displacement and the restoring force are continually in opposite directions;
acceleration a – displacement; and
force = - kx where the –ve stands for direction, the constant k is +ve
the one force – gravity – acts downwards;
the force is constant;
as the displacement increases the force remains constant in the same direction; and
acceleration is constant
ie ms-1

14. Periodic and approximates SHM
Periodic and Simple Harmonic Motion (SHM)
For these events discuss:
if they are periodic; and
if they approximate SHM.
Periodic and approximates SHM
The piston is periodic and approximates SHM at constant r.p.m.
The piston is driven i.e., given an impulse periodically to keep it going
The motion repeats over and over again in regular cycles
The crankshaft has constant angular velocity at constant r.p.m.
Piston
Crank shaft

15. Periodic but not SHM Periodic and Simple Harmonic Motion (SHM)
For these events discuss:
if they are periodic; and
if they approximate SHM.
Heart beat
Periodic but not SHM
A pulse beats at regular intervals.
Has no other SHM properties.
There is one main force – gravity – that acts downwards
The force is constant
As the displacement increases the force remains constant in the same direction
Acceleration is constant
Ski jumping
In free fall the motion is not periodic and hence cannot be SHM