simple harmonic motion level 1 physics. simple harmonic motion when a force causes back and forth...
Post on 02-Jan-2016
227 Views
Preview:
TRANSCRIPT
Simple Harmonic MotionLevel 1 Physics
Simple Harmonic MotionWhen a force causes back and forth motion that is directly proportionalto the displacement, it is said to be Simple Harmonic Motion
€
Fs α x
Hooke’s LawEasiest example of SHM would be Hooke’s Law
€
Fs α x
k = Constant of Proportionality Spring Constant Units: N/m
F = - kx Negative sign indicatesthat the force is in the opposite direction ofthe displacement
“F” is a Restoring Force – object comesback towards equilibrium
Hooke’s LawIn Hooke’s Law, you have to determine what force is stretching/compressingthe spring. Examples are;
€
Fs = kx
Fs = Fg → mg = kx
Fs = Fnet → ma = kx
Example
A load of 50 N attached to a spring hanging vertically stretches the spring 5.0 cm. The spring is now placed horizontally on a table and stretched 11.0 cm. What force is required to stretch the spring this amount?
k
k
kxFs
)05.0(501000 N/m
s
s
s
F
F
kxF
)11.0)(1000(
110 N
Hooke’s Law from a Graphical Point of View
x(m) Force(N)
0 0
0.1 12
0.2 24
0.3 36
0.4 48
0.5 60
0.6 72
graph x vs.F a of Slope
kx
Fk
kxF
s
sSuppose we had the following data:
Force vs. Displacement y = 120x + 1E-14
R2 = 1
0
10
20
30
40
50
60
70
80
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Displacement(Meters)
Fo
rce(
New
ton
s)
k =120 N/m
F vs. x again
Force vs. Displacement y = 120x + 1E-14
R2 = 1
0
10
20
30
40
50
60
70
80
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Displacement(Meters)
Fo
rce(
New
ton
s)
Area = Elastic Potential Energy
When you compress or stretcha spring, you must get something in return. In this case, the area under the curve is the energy.
Fx!
Energy is stored when a spring is stretched/compressed. This type ofenergy is classified as Elastic Potential Energy (EPE).
Elastic Potential EnergyThe graph of F vs. x for a spring will always producea linear line that has a positiveslope (IDEAL spring!). Thereforethe area under the curve will be a triangle.
€
At =1
2bh
At =1
2Fx
At =1
2kx( )x[ ]
At =1
2kx 2 ⇒ EPE =
1
2kx 2
€
At =1
2kx 2
Conservation of Energy in Springs
ExampleA slingshot consists of a light leather cup, containing a
stone, that is pulled back against 2 rubber bands. It takes a force of 30 N to stretch the bands 1.0 cm (a) What is the potential energy stored in the bands when a 50.0 g stone is placed in the cup and pulled back 0.20 m from the equilibrium position? (b) With what speed does it leave the slingshot?
v
vmvU
KUEEc
kkxUb
kkkxFa
s
sAB
s
s
22
2
)050.0(21
21
)
)20)(.(5.021)
)01.0(30) 3000 N/m
300 J
109.54 m/s
Springs are like Waves and Circles
The amplitude, A, of a wave is the same as the displacement ,x, of a spring. Both are in meters.
CREST
Trough
Equilibrium Line
Period, T, is the time for one revolution or in the case of springs the time for ONE COMPLETE oscillation (One crest and trough). Oscillations could also be called vibrations and cycles. In the wave above we have 1.75 cycles or waves or vibrations or oscillations.
Ts=sec/cycle. Let’s assume that the wave crosses the equilibrium line in one second intervals. T =3.5 seconds/1.75 cycles. T = 2 sec.
FrequencyThe FREQUENCY of a wave is the inverse of the
PERIOD. That means that the frequency is the #cycles per sec. The commonly used unit is HERTZ(HZ).
Tf
fT
Hzsccyc
fFrequency
scyc
sTPeriod
11
5.05.0sec5.3
75.1
seconds
cycles
275.1
5.3
cycles
seconds
PendulumsPendulums, like springs,
oscillate back and forth exhibiting simple harmonic behavior.
A shadow projector would show a pendulum moving in synchronization with a circle. Here, the angular amplitude is equal to the radius of a circle.
The Period of a Pendulum
g
lTpendulum 2
alityProportion ofConstant 4
4
2
2
22
g
lT
g
lT
ExampleA visitor to a lighthouse wishes to determine
the height of the tower. She ties a spool of thread to a small rock to make a simple pendulum, which she hangs down the center of a spiral staircase of the tower. The period of oscillation is 9.40 s. What is the height of the tower?
2
2
2
222
)141592.3(4
)8.9(4.9
4
4
2
gTl
g
lT
heightlg
lT
PP
P
L = Height = 21.93 m
Physics AppletTake a look at the following simulation:
UC Irvine Physics of Music Simple Harmonic Motion Applet Demonstrations
top related