Lecture 20

Physics 207: Lecture 20, Pg 1

Lecture 20Goals:Goals:

• Chapter 14Chapter 14Compare and contrast different systems with SHM.Understand energy conservation (transfer) in SHM. Understand the basic ideas of damping and resonance.

• Chapter 15Chapter 15 Understand pressure in liquids and gases Use Archimedes’ principle to understand buoyancy Understand the equation of continuity Use an ideal-fluid model to study fluid flow. Investigate the elastic deformation of solids and liquids

• AssignmentAssignment HW9, Due Wednesday, Apr. 7th Tuesday (after break!!!): Read all of Chapter 15


SHM So Far

The most general solution is x = A cos(t + )

where A = amplitude

= (angular) frequency

= phase constant

For SHM without friction,

The frequency does not depend on the amplitude ! This is true of all simple harmonic motion!

The oscillation occurs around the equilibrium point where the force is zero!

Energy is a constant, it transfers between potential and kinetic

mk


The “Simple” Pendulum

A pendulum is made by suspending a mass m at the end of a string of length L. Find the frequency of oscillation for small displacements.

Fy = may = T – mg cos() = mac = m vT2/L

Fx = max = -mg sin()

where x = L tan If small then x L and sin()

0° tan 0.00 = sin 0.00 = 0.00

5° tan 0.09 = sin 0.09 = 0.09

10° tan 0.17 = sin 0.17 = 0.17

15° tan 0.26 = 0.27

sin 0.26 = 0.26

L

m

mg

z

y

x

T


The “Simple” Pendulum

A pendulum is made by suspending a mass m at the end of a string of length L. Find the frequency of oscillation for small displacements.

Fy = may = T – mg cos() = mac = m vT2/L

Fx = max = -mg sin()

where x = L tan If small then x L and sin()

dx/dt = L d/dt

ax = d2x/dt2 = L d2/dt2

so ax = -g = L d2/ dt2 L d2/ dt2 + g = 0

and = cos(t + ) or = sin(t + )

with = (g/L)½

L

m

mg

z

y

x

T


SHM So Far

The most general solution is x(t) = A cos(t + )

where A = amplitude

= (angular) frequency = 2 f = 2/T

= phase constant

mk

Velocity: v(t) = -A sin(t + )

Acceleration: a(t) = -2A cos(t + )

L

gSimple Pendulum:

Hooke’s Law Spring:Spring constant

Inertia


SHM So Far

For SHM without friction

The frequency does not depend on the amplitude !

The oscillation occurs around the equilibrium point where the force is zero!

Mechanical Energy is constant, it transfers between potential and kinetic energies.


The shaker cart You stand inside a small cart attached to a heavy-duty spring, the

spring is compressed and released, and you shake back and forth, attempting to maintain your balance. Note that there is also a sandbag in the cart with you.

At the instant you pass through the equilibrium position of the spring, you drop the sandbag out of the cart onto the ground.

What effect does jettisoning the sandbag at the equilibrium position have on the amplitude of your oscillation?

It increases the amplitude.It decreases the amplitude.It has no effect on the amplitude.


The shaker cart Instead of dropping the sandbag as you pass through equilibrium, you

decide to drop the sandbag when the cart is at its maximum distance from equilibrium.

What effect does jettisoning the sandbag at the cart’s maximum distance from equilibrium have on the amplitude of your oscillation?

It increases the amplitude.

It decreases the amplitude.

It has no effect on the amplitude.


The shaker cart

What effect does jettisoning the sandbag at the cart’s maximum distance from equilibrium have on the maximum speed of the cart?

It increases the maximum speed.

It decreases the maximum speed.

It has no effect on the maximum speed.


What about Vertical Springs?

For a vertical spring, if y is measured from the equilibrium position

Recall: force of the spring is the negative derivative of this function:

This will be just like the horizontal case:

-ky = ma =

j j

k

m F= -ky

y = 0

U ky1

22

kydy

dUF NET

2

2

dtyd

m

Which has solution y(t) = A cos( t + ) km

where


Exercise Simple Harmonic Motion

A mass oscillates up & down on a spring. It’s position as a function of time is shown below. At which of the points shown does the mass have positive velocity and negative acceleration ?

Remember: velocity is slope and acceleration is the curvature

t

y(t)

(a)

(b)

(c)

y(t) = A cos( t + )

v(t) = -A sin( t + )

a(t) = -A cos( t + )


The “Torsional” Pendulum A pendulum is made by suspending a mass m at the end

of a string of length L. Find the frequency of oscillation for small displacements.

z = Iz = -mg sin() L

z ≈ mL2z ≈ -mg L

L (d2 /dt2) = -g

d2 /dt2 = (-g/L)

with (t)= 0 sin t or 0 cos t

and =(g/L)½ or if a true horizontal torsional pendulum

Iz = - with =(/I)½

L

m

mg

z

y

x

T


Exercise Simple Harmonic Motion

You are sitting on a swing. A friend gives you a small push and you start swinging back & forth with period T1.

Suppose you were standing on the swing rather than sitting. When given a small push you start swinging back & forth with period T2.

Which of the following is true recalling that = (g / L)½

(A) T1 = T2

(B) T1 > T2

(C) T1 < T2 T1 T2


Energy in SHM

For both the spring and the pendulum, we can derive the SHM solution using energy conservation.

The total energy (K + U) of a system undergoing SMH will always be constant!

This is not surprising since there are only conservative forces present, hence energy is conserved.

-A A0x

U

U

KE


SHM and quadratic potentials

SHM will occur whenever the potential is quadratic. For small oscillations this will be true: For example, the potential between

H atoms in an H2 molecule lookssomething like this:

-A A0x

U

U

KEU

x


See: http://hansmalab.physics.ucsb.edu

SHM and quadratic potentials

Curvature reflects the spring constant

or modulus (i.e., stress vs. strain or

force vs. displacement)

Measuring modular proteins with an AFM

U

x


What about Friction?A velocity dependent drag force (A model)

2

2

dtxd

mdtdxbkx

We can guess at a new solution.

)(cos expA )( )(2

ttxmbt

With,

22

2

22

mb

mb

mk

o

02

2

xm

k

dt

dx

m

b

dt

xd

and now 02 ≡ k / m

Note


What about Friction?

A damped exponential

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

A)(cos

2expA )( )( tt

mbtx mbo 2/if


Variations in the damping

Small damping time constant (m/b)

Low friction coefficient, b << 2m

Moderate damping time constant (m/b)

Moderate friction coefficient (b < 2m)


Damped Simple Harmonic Motion

A downward shift in the angular frequency There are three mathematically distinct regimes

22 )2/( mbo

mbo 2/ mbo 2/

underdamped critically damped overdamped

mbo 2/


Driven SHM with Resistance Apply a sinusoidal force, F0 cos (t), and now consider what A and b do,

2220

2

0

)()(

/

mb

mFA

b/m small

b/m middling

b large

tm

Fx

m

k

dt

dx

m

b

dt

xd cos 2

2

Not Zero!!!

stea

dy s

tate

am

plitu

de


Resonance-based DNA detection with nanoparticle probes

Change the mass of the cantilever & change the resonant frequency

Su et al., APL 82: 3562 (2003)


Exercise Resonant Motion

Consider the following set of pendulums all attached to the same string

D A

B

C

If I start bob D swinging which of the others will have the largest swing amplitude ?

(A) (B) (C)


Chapter 15, Fluids This is an actual photo of an iceberg, taken by a rig manager for

Global Marine Drilling in St. Johns, Newfoundland. The water was calm and the sun was almost directly overhead so that the diver


Fluids (Ch. 15)

At ordinary temperature, matter exists in one of three states Solid - has a shape and forms a

surface Liquid - has no shape but forms a

surface Gas - has no shape and forms no

surface What do we mean by “fluids”?

Fluids are “substances that flow”…. “substances that take the shape of the container”

Atoms and molecules are free to move.

No long range correlation between positions.


Fluids

An intrinsic parameter of a fluid Density

V

m

units :kg/m3 = 10-3 g/cm3

(water) = 1.000 x 103 kg/m3 = 1.000 g/cm3

(ice) = 0.917 x 103 kg/m3 = 0.917 g/cm3

(air) = 1.29 kg/m3 = 1.29 x 10-3 g/cm3

(Hg) = 13.6 x103 kg/m3 = 13.6 g/cm3


Fluids

nF ˆAp

A

n

Any force exerted by a fluid is perpendicular to a surface of contact, and is proportional to the area of that surface.

Force (a vector) in a fluid can be expressed in terms of pressure (a scalar) as:

A F

p Another parameter: Pressure


What is the SI unit of pressure?

A. PascalB. AtmosphereC. BernoulliD. YoungE. p.s.i.

Units : 1 N/m2 = 1 Pa (Pascal)1 bar = 105 Pa1 mbar = 102 Pa1 torr = 133.3 Pa

1 atm = 1.013 x105 Pa = 1013 mbar

= 760 Torr = 14.7 lb/ in2 (=PSI)


If the pressures were different, fluid would flow in the tube!

Pressure vs. Depth For a uniform fluid in an open

container pressure same at a given depth independent of the container

p(y)

y

Fluid level is the same everywhere in a connected container, assuming no surface forces

Why is this so? Why does the pressure below the surface depend only on depth if it is in equilibrium?

Imagine a tube that would connect two regions at the same depth.


Lecture 20

• AssignmentAssignment HW9, Due Wednesday, Apr. 7th Tuesday: Read all of Chapter 15

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Lecture 20

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