Ch. 10 Elasticity and Oscillations - Link to Unit 5 Link to Unit 5 Link to ...

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Nov 3, 2013 (3 years and 8 months ago)

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10.5 SHM

Concept Check


Oscillations

Which of the following is necessary to make an object oscillate?


1. a stable equilibrium

2. little or no friction

3. a disturbance

4. all of the above


Concept Check


Oscillations

Which of the following is necessary to make an object oscillate?


1. a stable equilibrium

2. little or no friction

3. a disturbance

4. all of the above

A stable equilibrium point is
needed. In addition, some sort of
disturbance is required to set the
object in motion (otherwise the
object would simply remain at the
equilibrium point). Finally, friction
should be small or absent,
otherwise the object doesn’t
oscillate but just returns to the
equilibrium point without
overshooting it.

Concept Check


Oscillations (2)

An object can oscillate around


1. any equilibrium point.

2. any stable equilibrium point.

3. certain stable equilibrium points.

4. any point, provided the forces exerted on it obey Hooke’s law.

5. any point.


Concept Check


Oscillations (2)

An object can oscillate around


1. any equilibrium point.

2. any stable equilibrium point.

3. certain stable equilibrium points.

4. any point, provided the forces exerted on it obey Hooke’s law.

5. any point.


When an object in stable equilibrium is
disturbed, it tends to return to the
equilibrium point. This is the basic
requirement for an oscillation. An object in
unstable equilibrium tends to move farther
away from equilibrium when it is
disturbed. Forces around any stable
equilibrium obey Hooke’s law, provided
the displacement from equilibrium is not
too large, so any stable equilibrium point
will do.

Concept Check


SHM

A mass attached to a spring oscillates back and forth as indicated in
the position vs. time plot below. At point
P
, the mass has








1. positive velocity and positive acceleration.

2. positive velocity and negative acceleration.

3. positive velocity and zero acceleration.

4. negative velocity and positive acceleration.

5. negative velocity and negative acceleration.

6. negative velocity and zero acceleration.

7. zero velocity but is accelerating (positively or negatively).


Concept Check


SHM

A mass attached to a spring oscillates back and forth as indicated in
the position vs. time plot below. At point
P
, the mass has








1. positive velocity and positive acceleration.

2. positive velocity and negative acceleration.

3. positive velocity and zero acceleration.

4. negative velocity and positive acceleration.

5. negative velocity and negative acceleration.

6. negative velocity and zero acceleration.

7. zero velocity but is accelerating (positively or negatively).


The velocity is
positive

because the slope of the
curve at point
P
is positive.
The acceleration is
negative

because the curve is
concave down at
P
.

Concept Check


SHM (2)

A mass suspended from a spring is oscillating up and down as
indicated. Consider two possibilities:

(i)

at some point during the oscillation the mass has zero velocity but is
accelerating (positively or negatively);

(ii)

at some point during the oscillation the mass has zero velocity and zero
acceleration.




1.

Both occur sometime during the oscillation.

2.

Neither occurs during the oscillation.

3.

Only
(i)

occurs.

4.

Only
(ii)

occurs.

Concept Check


SHM (2)

A mass suspended from a spring is oscillating up and down as
indicated. Consider two possibilities:

(i)

at some point during the oscillation the mass has zero velocity but is
accelerating (positively or negatively);

(ii)

at some point during the oscillation the mass has zero velocity and zero
acceleration.




1.

Both occur sometime during the oscillation.

2.

Neither occurs during the oscillation.

3.

Only
(i)

occurs.

4.

Only
(ii)

occurs.

The velocity is zero at the maxima and minima of the curve.
At these points, the curve is concave either up or down, and
so the particle is accelerating. The particle has zero
acceleration at the points of inflection of the curve, which
occur at those times
t
for which
x
= 0. At these points, the
curve has nonzero slope and so the velocity cannot be zero.

10.5 SHM

T
f
1

cycle
per

time

T
time
of
unit
per

cycles

f
(sec)
T


Hz
or

sec
cycles






f
Periodic Motion Terms



cos
x A t
 
 
t

x

A

-
A

T

¾
T

½
T

¼
T

10.5 SHM

Harmonic Motion

Click on above image to view the SHM java applet

f


2

T

2

10.5 SHM

Harmonic Motion

f


2

T

2

10.5 SHM

max
cos cos
x A x t
 
 
Displacement in SHM

A


P

Q

x
max


cos
A
x

A


P

Q

x
max


cos
A
x

O

O

10.5 SHM


r
v
T

Velocity in SHM

A


P

Q

x
max


cos
A
x

A


P

Q

x
max


cos
A
x


A
v

max
fA

2

v
max


v
x



sin
A
v
x


t
A


sin


O

O

10.5 SHM

Velocity in SHM

A


P

Q

x
max


cos
A
x

A


P

Q

x
max


cos
A
x

v
max



sin
A
v
x


O

O

2 2
y A x
 
sin
y
A


2 2
A x
A


2 2
1
x A
 
2 2
max
1
x
v v x A
  
y

10.5 SHM

Acceleration in SHM

A


P

Q

x
max


cos
A
x

A


P

Q

x
max


cos
A
x

v
max

2
c
v
a
r

O

O

cos
x c
a a

 
2
r


2
A


2
cos
A
 
 
2
cos
A t
 
 
2
x
a x

 
10.5 SHM

Displacement, Velocity, and Acceleration in SHM

t
A
x

cos

sin
x
dx
v A t
dt
 
  
2
cos
x
dv
a A t
dt
 
  
Java applet showing SHM of a mass
-
spring system

10.5 SHM

Displacement, Velocity, and Acceleration in SHM

Ranking Task


SHM Equations

Listed below are mathematical representations for objects undergoing simple harmonic
motions (SHM). In these expressions,
y

is in meters,
t

is in seconds, and the numerical
values have appropriate units.







Rank these mathematical representations of SHM, from greatest to least, on the basis of
the
period
, or time it takes for the object to complete one cycle of this motion.


Greatest








Least

1 ________ 2 ________ 3 ________ 4 ________ 5 ________ 6 ________

Or, it is not possible to compare these representations without knowing more. ________

Explain:


6sin3
y t

A

3sin6
y t

B

6cos3
y t

C



6sin 3 30
y t
  
D

10cos6
y t

E

10sin2
y t

F

Ranking Task


SHM Equations

Listed below are mathematical representations for objects undergoing simple harmonic
motions (SHM). In these expressions,
y

is in meters,
t

is in seconds, and the numerical
values have appropriate units.







Rank these mathematical representations of SHM, from greatest to least, on the basis of
the
period
, or time it takes for the object to complete one cycle of this motion.


Greatest








Least

1 ________ 2 ________ 3 ________ 4 ________ 5 ________ 6 ________

Or, it is not possible to compare these representations without knowing more. ________

Explain:
Period is the inverse of frequency. The angular frequency in each expression is
given by


in the following general expression for SHM:


6sin3
y t

A

3sin6
y t

B

6cos3
y t

C



6sin 3 30
y t
  
D

10cos6
y t

E

10sin2
y t

F

F

ACD

BE



cos
x A t
 
 
Ranking Task


SHM Equations

Listed below are mathematical representations for objects undergoing simple harmonic
motions (SHM). In these expressions,
y

is in meters,
t

is in seconds, and the numerical
values have appropriate units.







Rank these mathematical representations of SHM, from greatest to least, on the basis of
the
maximum velocity

of the object during one complete cycle of this motion.


Greatest








Least

1 ________ 2 ________ 3 ________ 4 ________ 5 ________ 6 ________

Or, it is not possible to compare these representations without knowing more. ________

Explain:


6sin3
y t

A

3sin6
y t

B

6cos3
y t

C



6sin 3 30
y t
  
D

10cos6
y t

E

10sin2
y t

F

Ranking Task


SHM Equations

Listed below are mathematical representations for objects undergoing simple harmonic
motions (SHM). In these expressions,
y

is in meters,
t

is in seconds, and the numerical
values have appropriate units.







Rank these mathematical representations of SHM, from greatest to least, on the basis of
the
maximum velocity

of the object during one complete cycle of this motion.


Greatest








Least

1 ________ 2 ________ 3 ________ 4 ________ 5 ________ 6 ________

Or, it is not possible to compare these representations without knowing more. ________

Explain:
Velocity of an object in SHM is given by the expression:

Maximum velocity occurs when sin = 1, or:


6sin3
y t

A

3sin6
y t

B

6cos3
y t

C



6sin 3 30
y t
  
D

10cos6
y t

E

10sin2
y t

F

E

F

ABCD



sin
v A t
  
 
max
v A

 
Ranking Task


SHM Equations

Listed below are mathematical representations for objects undergoing simple harmonic
motions (SHM). In these expressions,
y

is in meters,
t

is in seconds, and the numerical
values have appropriate units.







Rank these mathematical representations of SHM, from greatest to least, on the basis of
the
maximum acceleration

of the object during one complete cycle of this motion.


Greatest








Least

1 ________ 2 ________ 3 ________ 4 ________ 5 ________ 6 ________

Or, it is not possible to compare these representations without knowing more. ________

Explain:


6sin3
y t

A

3sin6
y t

B

6cos3
y t

C



6sin 3 30
y t
  
D

10cos6
y t

E

10sin2
y t

F

Ranking Task


SHM Equations

Listed below are mathematical representations for objects undergoing simple harmonic
motions (SHM). In these expressions,
y

is in meters,
t

is in seconds, and the numerical
values have appropriate units.







Rank these mathematical representations of SHM, from greatest to least, on the basis of
the
maximum acceleration

of the object during one complete cycle of this motion.


Greatest








Least

1 ________ 2 ________ 3 ________ 4 ________ 5 ________ 6 ________

Or, it is not possible to compare these representations without knowing more. ________

Explain:
Acceleration of an object in SHM is given by the expression:

Maximum velocity occurs when cos = 1, or:


6sin3
y t

A

3sin6
y t

B

6cos3
y t

C



6sin 3 30
y t
  
D

10cos6
y t

E

10sin2
y t

F

E

B

ACD



2
cos
a A t
  
 
2
max
a A

 
F

Concept Check


Oscillating Spring

An object hangs motionless from a spring. When the object is pulled
down, the sum of the elastic potential energy of the spring and the
gravitational potential energy of the object and Earth


1. increases.

2. stays the same.

3. decreases.


Concept Check


Oscillating Spring

An object hangs motionless from a spring. When the object is pulled
down, the sum of the elastic potential energy of the spring and the
gravitational potential energy of the object and Earth


1. increases.

2. stays the same.

3. decreases.




If released from its new position, the object accelerates upward and
passes the equilibrium point with nonzero velocity. The object has
therefore gained kinetic energy. The two forms of potential energy
present are elastic potential energy of the spring and gravitational
potential energy. Even though the latter decreases as the object is
pulled down, the sum of the two must increase for the object to be able
to gain kinetic energy.

10.6 Elastic Restoring Force

An Oscillating Spring

o
k
m

 
o
o
f


2

since

m
k
f
o

2
1

k
m
T
o

2

kx
ks
F
e




ma
F

x
m
k
a
x










2
for an object in SHM
x
a x


ma kx
 
Concept Check


Pendulum

A person swings on a swing. When the person sits still, the swing
oscillates back and forth at its natural frequency. If, instead,
two people

sit on the swing, the natural frequency of the swing is


1. greater.

2. the same.

3. smaller.

Concept Check


Pendulum

A person swings on a swing. When the person sits still, the swing
oscillates back and forth at its natural frequency. If, instead,
two people

sit on the swing, the natural frequency of the swing is


1. greater.

2. the same.

3. smaller.




Oscillations are an interplay between inertia and a restoring force. The
extra person doubles both the rotational inertia of the swing as well as
the restoring torque. The two effects cancel.

Concept Check


Pendulum

A person swings on a swing. When the person sits still, the swing
oscillates back and forth at its natural frequency. If, instead, the person
stands

on the swing, the natural frequency of the swing is


1. greater.

2. the same.

3. smaller.

Concept Check


Pendulum

A person swings on a swing. When the person sits still, the swing
oscillates back and forth at its natural frequency. If, instead, the person
stands

on the swing, the natural frequency of the swing is


1. greater.

2. the same.

3. smaller.




By standing up, the distance of the center of mass to the pivot point is
reduced. The restoring torque decreases linearly with this distance; the
rotational inertia as the square of it. Thus, the rotational inertia
decreases more and the period decreases.

10.7 The Pendulum


sin
mg
l


sin restoring force
F mg




nd
2 Law
T
F ma

sin
T
F
a g
m

  

L
l

If sin, then
T
g
a l
L
 
 
  
 
 
L
g
f
o

2
1

g
L
T
o

2

2
x
a x

 
L
g
o


f
o


2

l
L


In order to be SHM, , not sin
T
a
 

10.7 The Pendulum

What happens to the
period

of a simple pendulum

if we increase its
length

so that it is 4 times longer?




What happens to the
frequency

of a simple pendulum

if we increase its
length

so that it is 4 times longer?




What happens to the
period

of a simple pendulum if we increase its

mass

so that it is 4 times greater?


sin
mg
l
o
T L

Increasing L by a factor of 4 increases

T by a factor of or 2.

4
Period and frequency are inverse of each other, so if the period is doubled,
according to the previous problem, frequency will be halved.

The period of a simple pendulum is independent of mass. Therefore, the
period will be the same with a mass that is 4 times greater.

10.8 Damping, Forcing, and Resonance

Damping

Java applet showing Damped Harmonic motion

Click on above image to go to Hyperphysics explanation of damped harmonic motion

10.8 Damping, Forcing, and Resonance

Forced Oscillation and Resonance

Java applet showing Forced Oscillation and Resonance

University of Salford explanation of Forced Oscillation