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