Describes simple harmonic motion of weight- spring systems and pendulums. **More good stuff available at: www.wsautter.com and http://www.youtube.com/results?search_query=wnsautter&aq=f

1. Copyright Sautter 2015

2. 200
grams
Vibrating
Tuning fork
A weight on
a spring
A boy on
a swing
2

3. Simple Harmonic Motion
• Simple harmonic motion (SHM) is a repeated
motion of a particular frequency and period.
• The force causing the motion is in direct relationship
to the displacement of the body. (Hooke’s Law)
• The displacement, velocity, acceleration and force
characteristics are specific a various points in the
cycle for SHM.
• SHM can be understood in terms of the
displacement, velocity, acceleration and force
vectors related to circular motion.
• Recall that the displacement vector for circular
motion is the radius of the circular path. The
velocity vector is tangent to the circular path and
the acceleration vector always points towards the
center of the circle.
3

4. 400
grams
200
grams
F
O
R
C
E
(N)
ELONGATION (M)
Slope = spring constant
600
grams
Elongation of spring
4

5. Simple Harmonic Motion
• SHM motion can be represented as a vertical
view of circular motion. Using this concept, we
can see the variations in the vector lengths and
directions for displacement, velocity and
accelerations as those values for SHM.
• Use the following slide showing a mass on a
spring, vibrating in SHM to examine the
variations in these three vectors as the reference
circle rotates.
• See if you can decide which trig functions (sine,
cosine or tangent) govern each to the three
vectors in SHM
5

6. Displacement = +max
Velocity = 0
Acceleration = - max
Kinetic Energy = 0
Net Force = - max
Displacement = 0
Velocity = max
Acceleration = 0
Kinetic Energy = max
Net Force = 0
Top of cycle
Mid cycle
Bottom of cycle
Displacement = - max
Velocity = 0
Acceleration = + max
Kinetic Energy = 0
Net Force = + max
CLICK
HERE
6

7. The velocity vector (black)
is always directed tangentially
to the circular path.
The acceleration vector (red)
is always directed toward
the center of the circular path
7

8. Displacement Vector of Circular motion
& Displacement in SHM
Acceleration
Vector
8

9. Displacement vector
on Reference Circle
200
grams
200
grams
200
grams
Vertical
View
Simple Harmonic
Position
200
grams
y = +max
y = 0
y = -max
y = 0
9

10. Displacement vector
on Reference Circle
Vertical
View
Note that the vertical
view of the
displacement vector
is 0 at 00, 100 % upward
at 900, 0 at 1800, 100 %
downward at 2700
and finally 0 again at 3600
What trig function is
0 at 00, 1.0 (100%) at 900,
0 at 1800, -1.0 (100% and
pointing down) at 2700,
and 0 again at 3600
y = 0
00
y = -max 2700
y = 0
1800
y = +max 900
The SINE
y = Amp x sin θ
10

11. The velocity vector is
always tangent to the
circular path
11

12. The reference circle is
turned sideways
and viewed vertically.
This shows the velocity
vector of a body in
Simple
Harmonic Motion.
12

13. Velocity Vector of Circular motion
& Velocity in SHM
13

14. Velocity vector on
Reference Circle
200
grams
200
grams
200
grams
Vertical
View
Simple Harmonic
Position
200
grams
V = 0
V = + max
V = 0
V = -max
14

15. Note that the vertical
view of the
velocity vector
is 100 % upward at 00,
0 at 900, 100% downward
at 1800, 0 at 2700
and finally 100% again
at 3600
What trig function is
1.0 (100%) at 00, 0 at 900,
1.0 (100%) at 1800,
0 at 2700, and
1.0 again at 3600
Velocity vector on
Reference Circle
Vertical
View
V = 0
900
V = + max
00
V = 0
2700
V = -max
1800
The COSINE
V = Vmax x cos θ
15

16. Acceleration Vector of Circular motion
& Acceleration in SHM 16

17. Acceleration vector
on Reference Circle
200
grams
200
grams
200
grams
Vertical
View
Simple Harmonic
Position
200
grams
a = -max
a = 0
a = +max
a = 0
17

18. Acceleration vector
on Reference Circle
Vertical
View
a = -max, 900
a = 0
00
a = +max, 2700
a = 0
1800
Note that the vertical
view of the
acceleration vector
is 0 at 00, 100 % downward
at 900, 0 at 1800,
100 % upward at 2700
and finally 0 again at 3600
What trig function is
0 at 00, -1.0 (100%) at 900,
0 at 1800, +1.0 (100% and
pointing down) at 2700,
and 0 again at 3600
The - SINE
a = amax x ( -sin θ)
18

19. 200
grams
200
grams
200
grams
200
grams
200
grams
0 o
90 o
180 o
270 o
360 o
V = Velocity max x Cos 
Acc. = Acc. max x (-Sin )
Y = Amplitude x Sin 
19

20. y
t
Y = Amp x Sin 
Amp
dy/dt
t
dy/dt = Vmax Cos 
d2y/dt2
t
d2y/dt2 = Amax (-Sin )
Displacement
Velocity
Acceleration
20

21. 21
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