Physics 207: Lecture 19, Pg 1 Lecture 20Goals: Wrap-up Chapter 14 (oscillatory motion) Wrap-up Chapter 14 (oscillatory motion) Start discussion of Chapter

Physics 207: Lecture 19, Pg 1

Lecture 20Goals:Goals:

• Wrap-up Chapter 14 (oscillatory motion)Wrap-up Chapter 14 (oscillatory motion)

• Start discussion of Chapter 15 (fluids)Start discussion of Chapter 15 (fluids)

• AssignmentAssignment HW-8 due Tuesday, Nov 15 Monday: Read through Chapter 15


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

where A = amplitude

= angular frequency

= phase constant

SHM Solution...

km

-A A0(≡Xeq)


km

-A A0(≡Xeq)

T = 1 s

k

-1.5A 1.5A0(≡Xeq)

T is:

A)T > 1 s

B)T < 1 s

C) T=1 s

m


SHM Solution...

The mechanical energy is conserved:

U = ½ k x2 = ½ k A2 cos2(t + )

K = ½ m v2 = ½ k A2 sin2(t+)

U+K = ½ k A2

U~cos2K~sin2

E = ½ kA2


km

-A A0

Which mass would have the largest kinetic energy

while passing through equilibrium?

A)

2k2m

-A A0

B)

k

-2A 2A0

mC)


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.


Energy in SHM

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

-A A0x

U

U

KE

U = ½ k x2


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


What about Vertical Springs?

k

m

k

equilibrium

new

equilibrium

mg=k ΔL

ΔL

k

m

y=0ΔL

y

Fnet= -k (y+ ΔL)+mg=-ky Fnet =-ky=ma=m d2y/dt2

Which has the solution y(t) = A cos( 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()

Fx = max = -mg sin()

If small then x L and sin() dx/dt = L d/dt

ax = d2x/dt2 = L d2/dt2

so ax = -g = L d2/ dt2

and = cos(t + )

with = (g/L)½

T= 2π(L/g)½

L

m

mg

z

y

x

T


What about friction?

One way to include friction into the model is to assume velocity dependent drag.

Fdrag= -bdragv=-bdrag dx/dt

Fnet=-kx-bdrag dx/dt = m d2x/dt2

d2x/dt2=-(k/m)x –(bdrag/m) dx/dt a new differential

equation for x(t) !


Damped oscillations

x(t) = A exp(-bt/2m) cos (t + )

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

A

x(t)

t

For small drag (under-damped) one gets:


Driven oscillations, resonance

So far we have considered free oscillations.Oscillations can also be driven by an external force.

extext

osci

llatio

n am

plitu

de

natural frequency


Chapter 15, Fluids An actual photo of an iceberg


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.


Fluids

An intrinsic parameter of a fluid Density (mass per unit volume)

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

ρ=m/V


Fluids

Another parameter Pressure (force per unit area)

P=F/A

SI unit for pressure is 1 Pascal = 1 N/m2

1 atm = 1.013 x105 Pa = 1013 mbar

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

The atmospheric pressure at sea-level is


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.

Documents

Physics 207: Lecture 19, Pg 1 Lecture 20Goals: Wrap-up Chapter 14 (oscillatory motion) Wrap-up Chapter 14 (oscillatory motion) Start discussion of Chapter