Simple Harmonic Motion

Lecturer: Professor Stephen T. Thornton

Reading Quiz Which 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

Do demos

Oscillations

Cone inside loudspeaker Copyright © 2009 Pearson Education, Inc.

Car coil springs

Copyright © 2009 Pearson Education, Inc.

Oscillations of a Spring 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.

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Oscillations of a Spring

•

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, 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

A

Simple harmonic motion We can describe this motion mathematically quite easily:

x



A

cos 2 

t T

when

x



A

at

t

 0 We obtain same result for time

t

and

t + T

. Look at previous slide. Math gives same result:

x



A

cos 2 

t T



A

 

ft

 

A

cos 

t



A

cos(   ) Note:  

t

 0

Note  

t

 0  cos(   )

t

= 0

t

= 0

Simple Harmonic Motion 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:

m dt

2

dt

2   

kx

 0

with solutions of the form:

x



A

cos(   ) Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc.

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

x



A

cos(   )

v



dx dt

  

A

sin(   )

a a

  

dt

 2 2 

x dv dt

  2

A

cos(   )

Simple Harmonic Motion



dt

2   2

x

  0

k m x

 0  

Because

  2 

f



k m

,  if  2 

k m

then

Copyright © 2009 Pearson Education, Inc.

T f

  2 1 2  

k m m k T



f

1

Conceptual Quiz

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?

A) 0 B) A/2 C) A D) 2A E) 4A

Conceptual Quiz

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?

A) 0 B) A/2 C) A D) 2A E) 4A In the time interval T (the period), the mass goes through one complete oscillation back to the starting point. The distance it covers is A + A + A + A (4A).

Conceptual Quiz

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?

A) x = A B) x > 0 but x < A C) x = 0 D) x < 0 E) none of the above

Conceptual Quiz

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

and a = A) x = A B) x > 0 but x < A C) x = 0 D) x < 0 E) none of the above If both

v

and

a

were zero at the same time, the mass would be at rest and stay at rest!

Thus, there is

NO point

at which both

v

and

a

are both zero at the same time.

Follow-up: 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  

t

, so that the rate of circular rotation is constant.

x



A

cos  

A

cos( 

t

) 

A

cos 2 

t T

Oscillations of a Spring

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.

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

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Energy

E U

cos(   )

K

 1 2

mv

2  1 2

mA

2  2 2 sin (   )

U

 1 2

kx

2  1 2

kA

2 2 cos ( )

U

max  1 2

kA

2

K

max  1 2

mA

2  2  1 2

mA

2

k

   1 2

kA

2

E K

1 2

kA

2 2 cos (   )  1 2

kA

2 2 sin (

E

 1 2

kA

2   2 cos (  2 )  

E

 1 2

kA

2  1 2

mv

2  1 2

kx

2

E

is total mechanical energy =

K

+

U

.

E

will be conserved in this case. We are assuming frictionless motion.

)

E

Energy as a Function of Position in Simple Harmonic Motion

U

1 2

mv

2  1 2

kx

2  1 2

kA

2

Energy as a Function of Time in Simple Harmonic Motion

Look at simulations http://physics.bu.edu/~duffy/seme ster1/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/s 2 . Find the value of (

a

) the spring constant and (

b

) the mass.

Conceptual 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?

A) 1/4 N B) 1/2 N C) 1 N D) 2 N E) 4 N

Conceptual 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?

A) 1/4 N B) 1/2 N C) 1 N D) 2 N E) 4 N 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!!