Physics 1B03summer-Lecture 10

Today’s Lecture…

… will start at 10:30am (and end at regular time)

Physics 1B03summer-Lecture 10

Day of Wrath

Tuesday June 16

9:30 am – 11:30 am

CNH-104

30 MC Questions, Cumulative

Physics 1B03summer-Lecture 10

Wave MotionWave Motion

•Energy and power in sinusoidal waves

Physics 1B03summer-Lecture 10

Energy in Waves

- as waves propagate through a medium, they transport energy

eg: ship moving up and down on a lakeeg: feeling sound waves at a rock concert

- hence, we can talk about energy and the ‘rate of energy transfer’

Physics 1B03summer-Lecture 10

Energy and Power

)(amplitudePower Energy, 2

A stretched rope has energy/unit length:

dx

dsdm

For small A and large , we can ignore the difference between “ds”, “dx” :

dm = μ dx (μ = mass/unit length)

Physics 1B03summer-Lecture 10

The mass dm vibrates in simple harmonic motion. Its maximum kinetic energy is dKmax = ½(dm)vmax

2

= ½(dm)(ωA)2

dE = ½(dm) ω 2A2

22

21

length)(unit A

E

The average kinetic energy is half this maximum value, but there is also an equal amount of potential energy in the wave. The total energy (kinetic plus potential) is therefore:

To get the energy per unit length (or energy ‘density’), replace the mass dm with the mass per unit length :

Physics 1B03summer-Lecture 10

Power: Energy travels at the wave speed v,

So

waves on a string,

lengthEnergy

vP

vAP 2221

Both the energy density and the power transmitted are proportional to the square of the amplitude. This is a general property of sinusoidal waves.

Physics 1B03summer-Lecture 10

Example

A string for which μ=5.0x10-2 kg/m is under tension of 80.0 N. How much power must be supplied to the string to generate sinusoidal waves at a frequency of 60Hz and with an amplitude of 6.0 cm ?

Physics 1B03summer-Lecture 10

Example

A sinusoidal wave on a string is described by the equation:

y(x,t) = (0.15m)sin(0.80x-50t)

where x is in meters and t in seconds. If μ=12.0g/m, determine:

a) the speed of the wave

b) the speed of particles on the wave at any time

c) the wavelength

d) the frequency

e) the power transmitted to the wave

Physics 1B03summer-Lecture 10

Quiz

The sound waves from your 100-watt stereo causes windows across the street to vibrate with an amplitude of 1 mm. If you use a 400-watt amplifier, what sort of amplitude can you get from the windows?

A) 2mmB) 4mmC) 16 mm

Physics 1B03summer-Lecture 10

IntensityIntensity

I = Power per unit area

Unit: W / m2

(the area is measured perpendicular to the wave velocity)

Intensity ~ (amplitude)2

source

detectors (area A)

Physics 1B03summer-Lecture 10

How would the intensity depend on distance from the source for:

1) waves spreading out equally in all directions in space? (This is called an“isotropic” source, or a source of “spherical waves”.)

2) Waves spreading out on a two-dimensional surface, e.g., circular ripples from a stone dropped into water?

How would the amplitude depend on distance?

Question

Physics 1B03summer-Lecture 10

10 min rest

Physics 1B03summer-Lecture 10

Fluid Mechanics and DynamicsFluid Mechanics and Dynamics

• Pressure• Pascal’s Law

• Buoyancy• Bernoulli’s Equation (Fluid Dynamics)

Physics 1B03summer-Lecture 10

- Includes liquids and gases. No resistance to “shear” (changes in shape), in equilibrium.

- To describe mechanics of a continous fluid (instead of a discrete object), we use density, pressure instead of mass and force.

- Dynamics is approached from an energy perspective (Bernoulli’s equation—next lecture) .

FluidFluidss

Physics 1B03summer-Lecture 10

Density, (“rho”), is mass per unit volume (kg/m3).

Specific Gravity (“SG”) is the ratio: (density of substance)/(density of water), which is a pure number (no units).

DensitDensityy

Substance SG

water 1000 kg/m3 1 mercury 13600 kg/m3 13.6 air 1.29 kg/m3 0.00129 helium 0.18 kg/m3

0.00018

Physics 1B03summer-Lecture 10

P force per unit area

unit: 1 N/m2 = 1 pascal (Pa)

Also, 1 atmosphere (atm) = 101.3 kPa

PressurPressuree

Pressure is a scalar property of the fluid; the force is always exerted perpendicular to the surface in contact with the fluid.

Forces exerted by the fluid

Physics 1B03summer-Lecture 10

Pascal’s Law: Pressure in an enclosed fluid in equilibrium is the same everywhere, except for differences due to gravity.

Or, pressure changes are transmitted throughout a fluid in equilibrium without loss; there is no static friction in fluids.

push here

Pressure increases here as well

Physics 1B03summer-Lecture 10

Example: Example: How hard do you need to push to lift a cement truck (weight W = 200 kN)?

F1 = ?

piston, radius 5mm

piston, radius 100mm

w

Physics 1B03summer-Lecture 10

Pressure variation with Pressure variation with depthdepth

h

P1

P2

Pressure increases with depth, by an amount

P2 – P1 gh

(if and g are uniform).

Proof: Consider forces on a cylinder of fluid

Physics 1B03summer-Lecture 10

“Gauge Pressure” : pressure difference between a fluid and the surrounding atmosphere. It is equal to P2–P1.

Example: a tire gauge measures gauge pressure, and reads zero when the air inside the tire is at atmospheric pressure.

“Absolute Pressure” is the pressure compared to vacuum. Zero absolute pressure means a vacuum.

Example: the pressure on the surface of the earth.

Physics 1B03summer-Lecture 10

Example

At what depth in water is the pressure 1 atm higher than the pressure on the surface? That is, where is P=2atms ?

Physics 1B03summer-Lecture 10

Example

What is the difference in air pressure between the floor and the ceiling?

Physics 1B03summer-Lecture 10

Example

What is the total mass of air directly above a 1-metre square, from ground level all the way to outer space?

Approximately how thick is the atmosphere, assuming (incorrectly) that the air density is uniform?