Exam3Results - Physics and Astronomy at TAMUpeople.physics.tamu.edu/tyana/PHYS218/files/Lect_26_12-1-14_post.pdf · DampedOscillaons+ • Real-world systems have some dissipative

Exam 3 Results

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Approximate Grade Cutoffs Ø  75 -‐ 100 A Ø  55 – 74 B Ø  35 – 54 C Ø  25 – 34 D Ø  0 – 24 F

You can pick them up a/er class today or Wednesday T. S%egler 12/1/2014 Texas A&M University

Chapter 14

T. S%egler 12/1/2014 Texas A&M University

Today •  Simple pendulums •  Physical pendulums •  Ch 14 Examples

Start Ch 15 (if we get to it)

Last Time

•  Simple harmonic mo%on of a mass-‐spring system

•  Differen%al equa%ons and SHM

•  Oscilla%ons in terms of amplitude, period, frequency and angular frequency

•  Simple harmonic mo%on using energy

To double the total energy of a mass-‐spring system oscilla%ng in simple harmonic mo%on, the amplitude must increase by a factor of

A. 4.

B.

C. 2.

D.

E.

=2 1.414.

=4 2 1.189.

=2 2 2.828.

0

02

0

( ) cos( )( ) sin( )

( ) cos( )

x t A tv t A ta t A t

ω φ

ω ω φ

ω ω φ

= +

= − +

= − +

21

pend

ff T

mgd gI L

ω π

ω

=

=

= =

12kA2 = 1

2mv2 + 1

2kx2 = const.

Clicker QuesEon Review from last Monday


SHM and Pendulums

•  For the physical pendulum the CoM is a distance d from the pivot, then the torque at an angle θ is:

! = !d(mgsin" )

•  Using the small angle approxima%on we obtain:

! = !dmg" so I! ! "dmg"

•  Comparing to the SHM equa%on again gives:

2

2

1 22

d mgddt I

mgd mgd If TI I mgd

θα θ

ω ππ

⎛ ⎞= = −⎜ ⎟⎝ ⎠

⇒ = ⇔ = ⇔ =


SHM and Pendulums

•  The simple pendulum is a special case of the physical pendulum. •  All the mass is located at a point aTached to the end of string of length L.

Using this d = L and I = mL2

2

mgd mgL gI mL L

ω = = =

! =gL! f = 1

2"gL!T = 2" L

g


Wri%ng down Newton's Second Law in the x direc%on for some system results in the equa%on on the right. What is the oscilla%on frequency of this system?

a2 d2xdt2

= !b2xa) a/b b) a*b c) b/a

Prelecture: Physical Pendula Problem 1


Fx =max

Fx =md 2xdt2

d 2xdt2

+! 2x = 0

Prelecture: Physical Pendula Problem 2

A uniform s%ck and a mass on a string are used to make two pendula that have the same length. Which one swings with the longer period?

a) the mass on the string b) the s%ck c) can't tell without knowing how the masses compare


! =mgdI

=mgLmL2

=gL

T = 2!"

=4! 2Lg

! =mgdI

=mgL13mL2

=3gL

T = 2!"

=4! 2L3g

(a)

(b)

Checkpoint: Physical Pendula Problem 1

A simple pendulum is used as the %ming element in a clock as shown. An adjustment screw is used to make the pendulum shorter (longer) by moving the weight up (down) along the sha_ that connects it to the pivot. If the clock is running too fast, the weight needs to be moved a)Up b)Down


T = 2! Lg

Checkpoint: Physical Pendula Problem 2

A torsion pendulum is used as the %ming element in a clock as shown. The speed of the clock is adjusted by changing the distance of two small disks from the rota%on axis of the pendulum. If we adjust the disks so that they are closer to the rota%on axis, the clock runs a) faster b) slower

! ="I

f = 12#

"I

T = 2# I"

Angular SHM: κ = torsion constant


Checkpoint: Physical Pendula Problem 3 Consider the two pendula shown above. In Case 1 a s%ck of mass M is pivoted at one end and used as a pendulum. In Case 2 a point par%cle of mass M is aTached to the center of the same s%ck. In which case is the period of the pendulum the longest? a) Case 1 b) Case 2 c) Same


! =mgdI

T = 2!"

= 2! Imgd

T1 = 2!

13mL2

mg L2

= 2! 2L3g

T2 = 2!

13mL2 +m L

2!

"#

$

%&

2!

"

##

$

%

&&

2mg L2

= 2! 7L12g

Damped OscillaEons

•  Real-world systems have some dissipative forces that decrease the amplitude.

•  The decrease in amplitude is called damping and the motion is called damped oscillation.

•  The figure at the right illustrates an oscillator with a small amount of damping.

•  The mechanical energy of a damped oscillator decreases continuously.

•  When , the system is critically damped and if b is larger than this it is overdamped

•  A critically or overdamped oscillator returns to equilibrium without oscillating.

b = 2 km


Example Pendulum on Mars (14.48)

A certain simple pendulum has a period on the earth of 1.60 s. What is its period on the surface of Mars where gravity is gmars = 3.71 m/s2?

0

02

0

( ) cos( )( ) sin( )

( ) cos( )


ω φ

ω ω φ

ω ω φ

= +

= − +

= − +

21

pend

ff T

mgd gI L

ω π

ω

=

=

= =


A simple pendulum consists of a point mass suspended by a massless, unstretchable string.

If the mass is doubled while the length of the string remains the same, the period of the pendulum

A. becomes 4 %mes greater.

B. becomes twice as great.

C. becomes greater by a factor of .

D. remains unchanged.

E. decreases.

2

Clicker QuesEon

0

02

0

( ) cos( )( ) sin( )

( ) cos( )


ω φ

ω ω φ

ω ω φ

= +

= − +

= − +

21

pend

ff T

mgd gI L

ω π

ω

=

=

= =

12kA2 = 1

2mv2 + 1

2kx2 = const.


Example Physical pendulum (small angle approxima%on)(14.54)

A 1.80 kg monkey wrench is pivoted 0.250 m from its center of mass and allowed to swing as a physical pendulum. The period for small angle oscilla%ons is 0.940 s. a)  What is the moment of iner%a of the wrench about an axis through the pivot?

b)  If the wrench is ini%ally displaced 0.400 rad from its equilibrium posi%on, what is the angular speed of the wrench as it passes through the equilibrium posi%on?

0

02

0

( ) cos( )( ) sin( )

( ) cos( )


ω φ

ω ω φ

ω ω φ

= +

= − +

= − +

21

pend

ff T

mgd gI L

ω π

ω

=

=

= =


Example Physical pendulum (14.57)

The two pendulums shown each consist of a uniform solid ball of mass M supported by a rigid massless rod, but the ball for pendulum A is very small compared to that of pendulum B. Find the period of each pendulum for small displacements. Which ball takes longer to complete a swing?

0

02

0

( ) cos( )( ) sin( )

( ) cos( )


ω φ

ω ω φ

ω ω φ

= +

= − +

= − +

21

pend

ff T

mgd gI L

ω π

ω

=

=

= =


Example Pendulums and collisions(14.95) In the figure the upper ball is released from rest, collides with the sta%onary lower ball, and s%cks to it. The strings are both 50.0 cm long. The upper ball has mass 2.00 kg, and ini%ally 10 cm higher than the lower ball (mass 3.00 kg). Find the frequency and maximum angular displacement of the mo%on a_er the collision.

0

02

0

( ) cos( )( ) sin( )

( ) cos( )


ω φ

ω ω φ

ω ω φ

= +

= − +

= − +

21

pend

ff T

mgd gI L

ω π

ω

=

=

= =


Example Energy and momentum in SHM

A block of mass M is aTached to a horizontal spring with spring constant k and is moving in SHM. As it passes through its equilibrium point a lump of puTy of mass m is dropped from a small height and s%ck to it. Find the new amplitude and period.


21

pend

ff T

mgd gI L

ω π

ω

=

=

= =

12kA2 = 1

2mv2 + 1

2kx2 = const.

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Exam3Results - Physics and Astronomy at TAMUpeople.physics.tamu.edu/tyana/PHYS218/files/Lect_26_12-1-14_post.pdf · DampedOscillaons+ • Real-world systems have some dissipative