simple harmonic motion lecturer: professor stephen t. thornton

Simple Harmonic Motion

Lecturer: Professor Stephen T. Thornton

Reading QuizWhich one of the following does not represent simple harmonic motion?

A) Distribution of student exam grades.

B) Automobile car springs.

C) Loudspeaker cone.

D) A mass oscillating at the end of a spring.

Answer: A

Last Time

Bernoulli equation/principle

Applications of Bernoulli principle

Read remaining sections of Chapter.

Today

Oscillations

Simple harmonic motion

Periodic motion

Springs

Energy

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Oscillations

Cone inside loudspeaker Car coil springs

Do demos

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If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic. The mass and spring system is a useful model for a periodic system.

Oscillations of a Spring

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• Displacement is measured from the equilibrium point.

• Amplitude A is the maximum displacement.

• A cycle is a full to-and-fro motion.

• Period is the time required to complete one cycle.

• Frequency is the number of cycles completed per second.

Oscillations of a Spring

Oscillations, simple harmonic motion, periodic motion

Start with periodic motion:

T = period of one cycle of periodic motion

f = 1/T = frequency of motion

unit of period: second

unit of frequency: 1 cycle/s = 1 Hz (hertz)

Displaying Position Versus Time for Simple Harmonic Motion

Chart paper moving up

pen

Simple Harmonic Motion as a Sine or a Cosine

Note period and amplitude T A

Simple harmonic motionWe can describe this motion mathematically quite easily:

2cos when at 0x A t x A t

T

We obtain same result for time t and t + T. Look at previous slide.

Math gives same result:

2cos cos 2 cos cos( )x A t A ft A t A t

T

Note: cos at 0x A t

Note cos at 0x A t

t = 0

t = 0

cos( )x A t

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Any vibrating system where the restoring force is proportional to the negative of the displacement is in simple harmonic motion (SHM), and is often called a simple harmonic oscillator (SHO).

F = ma = - kx Newton’s second law:

with solutions of the form:

Simple Harmonic Motion

2

2

2

20

d xm kxdt

d x k xmdt

cos( )x A t

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The velocity and acceleration for simple harmonic motion can be found by differentiating the displacement:

Simple Harmonic Motion

2 22

2

cos( )

sin( )

cos( )

x A t

dxv A tdtd x dva A t

dtdta x

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Simple Harmonic Motion

Because then

1Tf

22

2 2

0

0 if

d x k xmdtk kx xm m

km

2 / ,f k m

122

kf mmTk

Conceptual QuizConceptual Quiz

A) 0A) 0

B) B) AA/2/2

C) C) AA

D) 2D) 2AA

E) 4E) 4AA

A mass on a spring in SHM has

amplitude A and period T. What

is the total distance traveled by

the mass during a time interval T?

Conceptual QuizConceptual Quiz

A) 0A) 0

B) B) AA/2/2

C) C) AA

D) 2D) 2AA

E) 4E) 4AA

A mass on a spring in SHM has

amplitude A and period T. What

is the total distance traveled by

the mass after a time interval T?

In the time interval time interval TT (the period), the mass goes

through one complete oscillationcomplete oscillation back to the starting

point. The distance it covers is The distance it covers is A + A + A + AA + A + A + A (4 (4AA).).

Conceptual QuizConceptual Quiz

A) x = A

B) x > 0 but x < A

C) x = 0

D) x < 0

E) none of the above

A mass on a spring in SHM has amplitude A and period T. At what point in the motion is v = 0 and a = 0 simultaneously?

Conceptual QuizConceptual Quiz

A) x = A

B) x > 0 but x < A

C) x = 0

D) x < 0

E) none of the above

A mass on a spring in SHM has amplitude A and period T. At what point in the motion is v = 0 and a = 0 simultaneously?

If both If both vv and and aa were zero at were zero at

the same time, the mass the same time, the mass

would be at rest and stay at would be at rest and stay at

rest!rest! Thus, there is NO NO

pointpoint at which both vv and aa

are both zero at the same

time.Follow-up:Follow-up: Where is acceleration a maximum? Where is acceleration a maximum?

Connection between uniform circular motion and simple harmonic motion.

There is a remarkable relationship between the two.

Do projected uniform circular motion demo.

Let , so that the rate of

circular rotation is constant.

2cos cos( ) cos

t

x A A t A tT

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If the spring is hung vertically, the only change is in the equilibrium position, which is at the point where the spring force equals the gravitational force.

Oscillations of a Spring

This is the new equilibrium point. The mass oscillates about this level.

Energy

2 2 2 2

2 2 2

2max

2 2 2 2max

cos( )

1 1sin ( )

2 21 1

cos ( )2 2

Note maximum values of and .

1

21 1 1

2 2 2

E K U x A t

K mv mA t

U kx kA t

K U

U kA

kK mA mA kA

m

2 2 2 2

2 2 2

2 2 2

1 1cos ( ) sin ( )

2 21

cos ( ) sin ( )21 1 1

2 2 2

E U K kA t kA t

E kA t t

E kA mv kx

E is total mechanical energy = K + U. E will be conserved in this case. We are assuming frictionless motion.

Energy as a Function of Position in Simple Harmonic Motion

2 2 21 1 1

2 2 2E K U mv kx kA

Energy as a Function of Time in Simple Harmonic Motion

Look at simulations

http://physics.bu.edu/~duffy/semester1/semester1.html

Simple harmonic motion

Spring Oscillation. A vertical spring with spring stiffness constant 305 N/m oscillates with an amplitude of 28.0 cm when 0.260 kg hangs from it. The mass passes through the equilibrium point (y = 0) with positive velocity at t = 0. (a) What equation describes this motion as a function of time? (b) At what times will the spring be longest and shortest?

Oscillating Mass. A mass resting on a horizontal, frictionless surface is attached to one end of a spring; the other end is fixed to a wall. It takes 3.6 J of work to compress the spring by 0.13 m. If the spring is compressed, and the mass is released from rest, it experiences a maximum acceleration of 15 m/s2. Find the value of (a) the spring constant and (b) the mass.

A spring can be stretched a distance of 60 cm

with an applied force of 1 N. If an identical

spring is connected in parallel with the first

spring, and both are pulled together, how

much force will be required to stretch this

parallel combination a distance of 60 cm?

A) 1/4 N

B) 1/2 N

C) 1 N

D) 2 N

E) 4 N

Conceptual QuizConceptual Quiz

A spring can be stretched a distance of 60 cm

with an applied force of 1 N. If an identical

spring is connected in parallel with the first

spring, and both are pulled together, how

much force will be required to stretch this

parallel combination a distance of 60 cm?

Each spring is still stretched 60 cm, so each spring requires 1 N of force. But because there are two springs, there must be a total of 2 N of force! Thus, the combination of two parallel springs behaves like a stronger spring!!

Conceptual QuizConceptual QuizA) 1/4 N

B) 1/2 N

C) 1 N

D) 2 N

E) 4 N