SIMPLE HARMONIC MOTION AND
SPRINGS
SIMPLE HARMONIC MOTION
Starts from a stable point or a rest point
When an object is disturbed, it has a restorative force which tries
to restore the object to its rest position
 Generally, the force (and therefore) acceleration is proportional to displacement
This results in a back and forth motion that continues indefinitely
SPRING MOTION
When a spring is stretched and let go, it undergoes a longitudinal motion that is one
type of simple harmonic motion.
Restorative Force
Fspring  kx
k – spring constant (N/m)
x – displacement (m)
Why would the negative sign be present?
FOR THE PURPOSES OF SHM…
We consider an ideal spring…
 An ideal spring has no internal or external friction acting upon it
 In a practical (unrealistic) sense, this means that spring keeps oscillating forever
IF WE WERE TO EXAMINE THE MOTION WRT TIME
What does this look like?
Hint: The answer is not
fun. As much fun as it is,
that’s not what I’m looking
for.
What is a sine or cosine
function based on?
LET’S SEE HOW A UNIT CIRCLE RELATES TO SHM
https://www.geogebratube.org/student/m87292
How does SHM relate to the unit circle? Observe!
 Maximum speed through the rest point
 Stopped at the amplitude
 f of motion on a point of the circle = f of vibration in SHM
 They two motions are in phase
 Radius of the circle = amplitude
HOW CAN WE RELATE UCM TO SHM?
What do we know about things moving in a circle?
Derive
1
f 
2
k
m
m
T  2
k
SHM ISN’T REAL LIFE SO LET’S LOOK AT DHM
Damped Harmonic Motion
Periodic or repeated motion where amplitude decreases with time
There are no perfect springs!!!
DAMPING OF CAR SHOCKS
Damping for a car’s shocks:
- 0.7 is ideal in this case
-Overdamping is preferable to
underdamping
- Why is this so?
TOTAL ENERGY IN SHM
Energy is conserved!
Therefore, total energy remains constant
What is types of energy are being exchanged?
- Kinetic for Elastic potential and back
Therefore,
𝐸𝑇 = 𝐸𝑘 + 𝐸𝑒
𝐸𝑇 =
1
𝑚𝑣 2
2
1
2
+ 𝑘𝑥 2
EXAMPLE
A pendulum is disturbed from rest and is released from an amplitude of 15cm. If the
pendulum has a mass of 45g and a spring constant of 26N/m, what will the period
of the oscillation be?
EXAMPLE
A spring (k=20N/m) is compressed 30cm by a
ball (m= 100g) and fired upwards. How fast
will the object be moving after it has a vertical
displacement of 20cm after it leaves the
spring?
26. A 0.20 kg mass is hung from a vertical spring
of force constant 55N/m. When the spring is
released from its unstretched equilibrium position,
the mass is allowed to fall. Use the conservation of
energy to determine:
a) the speed of the mass after it falls 1.5cm.
b) the distance the mass will fall before reversing
direction
EXAMPLE
A 35kg child is bouncing on a pogo stick, if the spring constant is 4945N/m and it is
compressed by 25cm, how high will the child bounce?
QUESTION # 16
#18-20
Q # 13
REVIEW VIDEO
https://www.youtube.com/watch?v=VnGkoMoUkgI
4.5 Pg 217 Q 23 -26
Pg 219 Q12-14
P9 218 Q3-10
Review Pg 225 Q1-9,16
Pg 226 Q1,2,4,9-15,17,19 22, 24,25