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Describing Describing Periodic Periodic MotionMotion
AP PhysicsAP Physics
Hooke’s Law
sF k x
Restoring Force
The force exerted by a spring is a restoring force: it always opposes any displacement from equilibrium
Elastic Potential Energy
Work done is the area under the force vs. displacement graph
The area in this case can be found without calculus
Elastic Potential Energy
21
2ElasticU k x
Periodic Motion
Any motion which repeats itself is periodic. The time it takes to compete a cycle is the period of the system.
Examples: Perfect Bouncy Ball, Pendulum, Mass on a spring, spinning object
Example: Mass on Spring
Harmonic Motion
If a linear restoring force restrains the motion of an object, then the periodic motion is called simple harmonic motion
The system is called a Simple Harmonic Oscillator (SHO)
Harmonic Motion
Harmonic motion can be mathematically described by a sine function.
( ) sin( ) oy t A t y
Energy Conservation
If no energy is lost, a mass on a spring will remain in motion forever.
Sacred Tenant of Physics: The total energy of the system will be conserved!
constantKE U
Energy Conservation
21
2totalE kA
2 2 21 1 1
2 2 2mv kx kA
Example
A 1 kg. mass is attached to 25 N/m spring, stretched 10 cm from equilibrium and then released.
• What is the energy stored in the system before being released?
• What is the maximum velocity of the mass?
• What is the velocity when the mass is at x=5 cm?
Circular Motion
Simple Harmonic Motion can be compared with circular motion.
Demo
Derive the period of the system
Finding the Period
maxmax
2 2ma
m
x
ax
[ 1 ]
[ 2 ]
Solve [2] for v then sub into
2 2
1 1
2 2[1]
2
d A Av T
t T v
mv kA
mT
k
Period and FrequencyPeriod and Frequency
2
1
mT
k
fT
Angular FrequencyAngular Frequency
2k
fm
Mathematical ModelMathematical Model
Amplitude
Angular frequency
Equilibrium position
phase shift
( ) cos( )
o
o
A
x
x t A t x
Example 2Example 2
Write an equation for the position of a 0.3 kg. mass on a 100 N/m spring that is stretched from it’s equilibrium position of 15 cm to 18 cm then released.
• Find the period of the system, T
• Determine the angular frequency,
• Determine the Amplitude, A
• x(t) = Acos(t)+xo.
Example 3Example 3
The position function of a 100 g. mass is given by
( ) 0.12cos(2.8 ) 0.3x t t
Determine the following:
min max max max, , , , , ,f T k x x v a
Example 3 Solutions
1
0 2
2
max 0
min 0
22.240.12
2.80.446
0.3: use /
0.10.784
0.42
0.42
TA
f Tx
k k mm
k mx x A
x x A
Example 3 Solutions
max
max max2 2
max
max2
max
Use energy to find v
1 1
2 2 0.94 m/s
/ 0.336 m/s
total
F kA maE kA mv kA
am
v k m A
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