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 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 lake eg: 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

Energy, Power



(amplitude )

2 A stretched rope has energy/unit length:

dm dx ds

For small

A

and large l , 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

dK

max = ½(

dm

)

v

max 2 = ½(

dm

)(

ωA

) 2 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:

dE

= ½(

dm

)

ω

2

A

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

E

(unit length)  1 2  2

A

2 Physics 1B03summer-Lecture 10

Power: Energy travels at the wave speed v, So

P

   Energy length   

v

waves on a string,

P

 1 2  2

A

2

v

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) b) c) d) e) the speed of the wave the speed of particles on the wave at any time the wavelength the frequency 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) 2mm B) 4mm C) 16 mm Physics 1B03summer-Lecture 10

Intensity

I = Power per unit area Unit: W / m 2 Intensity ~ (amplitude) 2 detectors (area A)

(the area is measured perpendicular to the wave velocity)

source Physics 1B03summer-Lecture 10

Question 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?

Physics 1B03summer-Lecture 10

10 min rest

Physics 1B03summer-Lecture 10

Fluid Mechanics and Dynamics

• Pressure • Pascal’s Law • Buoyancy • Bernoulli’s Equation (Fluid Dynamics) Physics 1B03summer-Lecture 10

Fluids

- 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) .

Physics 1B03summer-Lecture 10

Density

Density, r (“rho”), is mass per unit volume (kg/m 3 ).

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

Substance

water mercury air helium

r 1000 kg/m 3 13600 kg/m 3 1.29 kg/m 3 0.18 kg/m 3 SG 1 13.6

0.00129

0.00018

Physics 1B03summer-Lecture 10

Pressure

P  force per unit area unit: 1 N/m 2 = 1 pascal (Pa) Also, 1 atmosphere (atm) = 101.3 kPa 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:

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

w F 1 = ?

piston, radius 5mm piston, radius 100mm

Physics 1B03summer-Lecture 10

Pressure variation with depth

Pressure increases with depth, by an amount P 2 – P 1

 r

gh (if

r

and g are uniform).

Proof: Consider forces on a cylinder of fluid h P 1 P 2

Physics 1B03summer-Lecture 10

“Gauge Pressure” : pressure difference between a fluid and the surrounding atmosphere. It is equal to P 2 –P 1 .

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?

Physics 1B03summer-Lecture 10