Describing Periodic Motion AP Physics. Hooke’s Law.
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Transcript of Describing Periodic Motion AP Physics. Hooke’s Law.
![Page 1: Describing Periodic Motion AP Physics. Hooke’s Law.](https://reader036.fdocuments.in/reader036/viewer/2022070409/56649e755503460f94b76163/html5/thumbnails/1.jpg)
Describing Describing Periodic Periodic MotionMotion
AP PhysicsAP Physics
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Hooke’s Law
sF k x
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Restoring Force
The force exerted by a spring is a restoring force: it always opposes any displacement from equilibrium
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Elastic Potential Energy
Work done is the area under the force vs. displacement graph
The area in this case can be found without calculus
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Elastic Potential Energy
21
2ElasticU k x
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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
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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)
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Harmonic Motion
Harmonic motion can be mathematically described by a sine function.
( ) sin( ) oy t A t y
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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
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Energy Conservation
21
2totalE kA
2 2 21 1 1
2 2 2mv kx kA
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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?
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Circular Motion
Simple Harmonic Motion can be compared with circular motion.
Demo
Derive the period of the system
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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
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Period and FrequencyPeriod and Frequency
2
1
mT
k
fT
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Angular FrequencyAngular Frequency
2k
fm
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Mathematical ModelMathematical Model
Amplitude
Angular frequency
Equilibrium position
phase shift
( ) cos( )
o
o
A
x
x t A t x
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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.
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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
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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
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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