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Simple Harmonic Motion Test

Tuesday 11/7

Chapter 11

Vibrations and Waves

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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.

We assume that the surface is frictionless. There

is a point where the spring is neither stretched

nor compressed; this is the equilibrium position.

We measure displacement from that point (x = 0

on the previous figure).

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•Period is the time required to

complete one cycle

• Displacement is measured from

the equilibrium point

•Amplitude is the maximum

displacement

•A cycle is a full to-and-fro motion;

this figure shows half a cycle

•Frequency is the number of cycles

completed per second

The force exerted by the spring depends on the

displacement:

• The minus sign on the force indicates that it is a

restoring force – it is directed to restore the mass

to its equilibrium position.

• k is the spring constant

• The force is not constant, so the acceleration is

not constant either

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Example

The spring constant of the spring

is 320 N/m and the bar indicator

extends 2.0 cm. What force does the

air in the tire apply to the spring?

(320 / )( 0.02 )

6.4

F kx

F N m m

F N

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.

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A spring has a length of 15.4 cm and is hanging vertically from a

support point above. A weight of 0.200 kg is then attached to the

spring, causing it to extend to a length of 28.6 cm. What is the

value of the spring constant? How much force is then needed to

lift this weight 4.6 cm from that position?

2

.286 .154 0.132

(0.200 )(10 / ) 2.0

2.0 (.132)

15.2 /

(15.2 / )(.046 )

0.7

x m

F kg m s N

N k

k N m

F N m m

F N

(.60)(20 ) 12

12 (50 / )

.24 24

244.8sec

5.0 /

s N

s

f F N N

f kx

N N m x

x m cm

cmtime

cm s

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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.

We already know that the potential energy of a

spring is given by:

The total mechanical energy of a spring system is:

The total mechanical energy will be conserved,

as we are assuming the system is frictionless.

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If the mass is at the limits of its

motion, the energy is all potential.

If the mass is at the equilibrium point,

the energy is all kinetic.

21The total energy is, therefore:

2kA

2 2 2

The energy equation for the system is

1 1 1+ =

2 2 2mv kx kA

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2 2

2 2

22

2 2

1 1

2 2

(10 / )(.1 )

(.01 )

3.2 /

kx mv

kx mv

kxv

m

kx N m mv

m kg

v m s

Assignment Read pg. 292-297

Do pg. 316-317

Questions

#2,5

Problems

#1,3,5,13

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Simple Harmonic Motion Test

Tuesday 11/7

Spring virtual lab

phet simulation

lab is on the class website

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Simple Harmonic Motion Test

Tuesday 11/7

What is the value of the spring constant of a spring

that is stretched a distance of 0.5 m if the restoring

force is 24 N?

a) 12 N/m

b) 18 N/m

c) 24 N/m

d) 48 N/m

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An object in simple harmonic motion is observed to move

between a maximum position and a minimum position. The

minimum time that elapses between the object being at its

maximum position and when it returns to that maximum

position is equal to which of the following parameters?

a) frequency

b) angular frequency

c) period

d) amplitude

A block is attached to the end of a spring. The block is then displaced

from its equilibrium position and released. Subsequently, the

block moves back and forth on a frictionless surface without any

losses due to friction. Which one of the following statements

concerning the total mechanical energy of the block-spring system

this situation is true?

a) The total mechanical energy is dependent on the maximum

displacement during the motion.

b) The total mechanical energy is at its maximum when the block is

at its equilibrium position.

c) The total mechanical energy is constant as the block moves back

and forth.

d) The total mechanical energy is only dependent on the spring

constant and the mass of the block.

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•Period is the time (seconds) required to complete one

cycle

•Frequency is the number of cycles completed per

second and is the reciprocal of the period

•The period of a spring system can be found using

•Frequency is measured in Hertz (Hz)

The Period and Sinusoidal Nature of SHM

max

2

max

2 f

v A

a A

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The frequency of motion is 1.0 KHz and the

amplitude is 0.20 mm.

(a)What is the maximum speed of the diaphragm?

(b)Where in the motion does this maximum speed

occur?

max

max

max

) 2

(2 )

(.0002 )(6.28)(1000 )

1.3 /

0

a f

v A A f

v m Hz

v m s

occurs

at

x

The displacement of an object is described by the following

equation, where x is in meters and t is in seconds:

x = (0.30m) cos (8.0 t)

Determine the oscillating object’s (a) amplitude, (b)

frequency, (c) period, (d) max speed, and (e) max acceleration

max

2 2

max

) 0.30

)8 6.28

1.3

1) .77sec

) (.3)(8) 2.4 /

) (.3)(8 ) 19.2 /

a amp m

b f

f Hz

c Tf

d v m s

e a m s

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The Simple Pendulum

A simple pendulum consists of a mass at the

end of a lightweight cord. We assume that

the cord does not stretch, and that its mass

is negligible.

The Simple Pendulum

In order to be in SHM, the restoring

force must be proportional to the

negative of the displacement. Here we

have:

which is proportional to sin θ and not to θ itself.

However, if the angle is small,

sin θ ≈ θ.

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The Simple Pendulum

Therefore, for small angles, we have:

where

The Simple Pendulum

So, as long as the cord can be

considered massless and the

amplitude is small, the period

does not depend on the mass.

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Determine the length of a simple pendulum that will

swing back and forth in simple harmonic motion with

a period of 1.00 s.

2

1 6.2810

.159210

.0253610

0.254

LT

g

L

L

L

L m

Pendulums and Energy Conservation

Energy goes back and forth between KE and PE.

At highest point, all energy is PE.

As it drops, PE goes to KE.

At the bottom , energy is all KE.

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Pendulum Energy ½mvmax

2 = mgh For minimum and maximum points of swing

A mass of 1.4kg is attached to a 3.2m long string to

make a simple pendulum.

a)What is the period of the pendulum?

b)If the pendulum is pulled back to an angle of 15o and

released, what is the maximum speed of the

pendulum?

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Assignment Do pg. 317-318

Problems

#9,16,21,24,28,30,32

Simple Harmonic Motion Test

Tuesday 11/7

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Pendulum virtual lab

phet simulation

lab is on the class website

due tomorrow

Simple Harmonic Motion Test

Tuesday 11/7

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Which one of the following units is used for frequency?

a) ohm

b) second

c) farad

d) hertz

Which one of the following statements concerning the total

mechanical energy of a harmonic oscillator at a particular

point in its motion is true?

a) The total mechanical energy depends on the acceleration at

that point.

b) The total mechanical energy depends on the velocity at that

point.

c) The total mechanical energy depends on the position of that

point.

d) The total mechanical energy does not vary during the motion

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A simple pendulum consists of a ball of mass m suspended

from the ceiling using a string of length L. The ball is

displaced from its equilibrium position by a small angle and

released. Which one of the following statements concerning

this situation is correct?

a) If the mass were increased, the period of the pendulum

would increase.

b) The frequency of the pendulum does not depend on the

acceleration due to gravity.

c) If the length of the pendulum were increased, the period of

the pendulum would increase.

d) The period of the pendulum does not depend on the length

of the pendulum.

A block of mass M is attached to one end of a spring that has a spring

constant k. The other end of the spring is attached to a wall. The block

is free to slide on a frictionless floor. The block is displaced from the

position where the spring is neither stretched nor compressed and

released. It is observed to oscillate with a frequency f. Which one of the

following actions would increase the frequency of the motion?

a) Decrease the mass of the block.

b) Increase the length of the spring.

c) Reduce the spring constant.

d) Reduce the distance that the spring is

initially stretched.

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In simple harmonic motion, an object oscillated

with a constant amplitude.

In reality, friction or some other energy dissipating

mechanism is always present and the amplitude

decreases as time passes.

This is referred to as damped harmonic motion.

Damped Harmonic Motion

However, if the damping is large, it

no longer resembles SHM at all.

A: underdamping: there are a few small oscillations before the

oscillator comes to rest.

B: critical damping: this is the fastest way to get to equilibrium.

C: overdamping: the system is slowed so much that it takes a

long time to get to equilibrium.

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1)simple harmonic motion

2&3) underdamped

4)critically damped

5) overdamped

Damped Harmonic Motion

There are systems where damping is unwanted, such as

clocks and watches.

Then there are systems in which it is wanted, and often needs

to be as close to critical damping as possible, such as

automobile shock absorbers and earthquake protection for

buildings.

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Forced Vibrations; Resonance Forced vibrations occur when there is a periodic driving force. This force may

or may not have the same period as the natural frequency of the system.

RESONANCE

Resonance is the condition in which a time-dependent force can transmit

large amounts of energy to an oscillating object, leading to a large amplitude

motion.

Resonance occurs when the frequency of the force

matches a natural frequency at which the object will

oscillate.

Forced Vibrations; Resonance

The sharpness of the

resonant peak depends

on the damping. If the

damping is small (A), it

can be quite sharp; if the

damping is larger (B), it

is less sharp.

Like damping, resonance can be wanted or unwanted.

Musical instruments and TV/radio receivers depend on it.

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When a force is applied to an oscillating system at all times,

the result is driven harmonic motion.

Here, the driving force has the same frequency as the

spring system and always points in the direction of the

object’s velocity.

Assignment Damped Harmonic Motion

assignment on the class

website