Simple Harmonic Motion

1
Simple Harmonic Motion

• Motion which is reoccurring
• Mass on spring
• Horizontal
• Vertical
• Simple Pendulum

2
S.H.M. Terminology

• Period (T) time for one complete oscillation
  (cycle)
• Sec/cycle
• Frequency (f) number of cycles in a given time.
• Cycles/sec (Hertz) (Hz)
• f 1/T
• Amplitude (A) Magnitude of the displacement
  relative to equilibrium or center position.

3
Fundamental of SHM
Frestore a -x
4
Mass - Spring System

• See IP demo

5
Derive Period (T) Equation
T ?
T vs Mass?
T vs Amplitude?
T vs k (spring constant)?
T vs gravity?
6
Mass - Spring System
T ind A
T ind g
7
Period (T) of Mass/Spring System
8
SHM SPRING-MASS
-A
A
0
Net force is attempting to return mass to
Equilibrium Position
A max displacement in x direction
- A max displacement in -x direction
9
SHM Kinematics
POSITION
0 p/2
p 3/2p
2p

VELOCITY
-
0 p/2
p 3/2p
2p

ACCEL
-
0 p/2
p 3/2p
2p
10
SHM Kinematics
POSITION
0 p/2
p 3/2p
2p

VELOCITY
-
0 p/2
p 3/2p
2p

ACCEL
-
0 p/2
p 3/2p
2p
11
SHM SPRING-MASS
-A
A
0
When mass to right
Fnet to the left
Frestore a -x
Frestore -kx
12
SHM SPRING-MASS
0
-A
A
When mass to left
Fnet to the right
Frestore a -x
Frestore -kx
13
Fundamental of SHM
Frestore a -x
Frestore -kx
k spring constant
14
SHM SPRING-MASS
0
-A
A
SOLVE FOR VELOCITY AT ANY POINT
15
SHM SPRING-MASS
0
-A
A
When mass _at_ X 0
Fnet 0
Speed is maximum
16
PE spring 1/2 kx2
PE spring 1/2 kA2
at x 0, KE mass PE spring
1/2 mvmax2 1/2 kA2
vmax2 (k/m)A2
17
Mass/Spring vs. Circular Motion

• What is Equal?
• Vx
• ax
• T
• A R

18
Relate mass-spring to circular motion
Speed distance / time
V C/T
T C / V
19
T 2pA
T 2p
20
Period (T) of Mass/Spring System
21
Simple Pendulum

• See IP demo

22
Derive Period (T) of Pendulum
T ?
T vs Mass?
T vs Amplitude?
T vs Length (L) ?
T vs gravity?
23
PERIOD (T) of Pendulum
T ind m
T ind Q
(for small angles)
24
Period (T) of Simple Pendulum
25
Simple Pendulum
v 0
Fc 0
sin q A/L
Wrad T
q
L
Net F Wtan -mgsin q
tan
Net F a sin q, not q
T
A
(Not true SHM)
Wrad
If q small, sin q q
q
Wtan
W
Net F mgA/L
rad
26
Relate pendulum to circular motion
Speed distance / time
V C / T
T C / V
27
Net F is Center Seeking F
Fc Net F
Fc mv2/R
Net F mgA/L
mv2/R mgA/L
(RA)
v2 (g/L)A2
28
T 2pA
T 2p
29
Period (T) of Simple Pendulum
30
Regents PhysicsHW 16 S.H.M.
REGENTS PHYSICS NAME _____________________ MR.
WHITMAN DATE _____________________

• CHAPTER 6.2
• PROBLEMS 1-6
• Due Mon., Day 1, Jan 23
• Exam on SHM Mon., Day 1, Jan 23

31
H.W. 12 14-6
32
H.W. 12 14-1