Physics 122 Concept Summary
Last Updated May 28, 2011
Note: Although I have done my best to check for typos and list the formulas correctly, you should verify the formulas are correct before using them. Make sure you know what all the variables represent in any particular formula. Some letters are used in different formulas from different chapters and may represent different things. - Dr. Nazareth
Fluids and Elasticity (Chapter 15)
- Mass Density (15.1)
- = m/VWhere m = mass and V = volume
- Pressure (15.2)
- P = F/A
- Where F = force of fluid acting perpendicular to surface and A = area of that surface.
- Pressure Gauges (15.2)
- Gauge pressure Pg = P - Patm
- This formula describes the pressure gauge (measuring device), not in what you are measuring (e.g., the gauge pressure at two different points in the water pipes of a house)
- P = the absolute pressure (as in, not relative to atmospheric pressure)
- Pressure and Depth in a Static Fluid (15.3)
- P2 = P1 + gh
- P2 is at the deeper point in the static fluid
- P1 is at the shallower point in the static fluid
- Pressure same at all points on horizontal line in connected static fluid
- Fluid rises to same height in all open regions of container
- Pascal’s Principle (15.3)
- Any change in pressure applied to a completely enclosed fluid is transmitted undiminished to all parts of the fluid and enclosing walls.
- Hydraulic lift (15.3)
- Archimedes’ Principle (15.4-5)
- Any fluid applies a buoyancy force to an object that is partially or completely submerged in it; the magnitude of the buoyancy force equals the weight of the fluid that the object displaces.
- FB = Wfluid displaced = ρfVfg
- The volume of the fluid displaced is equal to the volume of the submerged part of the object that is immersed in the fluid.
- Be careful to figure out how much of the volume is in the fluid (“below water”) and how much is above the fluid (“above water”).
- Equation of Continuity (15.6)
- Mass flow rate (Av) has the same value for every position along a tube that has a single entry point and a single exit point for fluid flow.
- 1A1v1 = 2A2v2
- = fluid density; A = cross-sectional area of tube; v = fluid speed
- Usually we assume an incompressible fluid (liquid), so 1 = 2 = which gives us
- A1v1 = A2v2(Equation of Continuity)
- Volume Flow RateQ = vA
- Bernoulli’s Principle (15.7-8)A statement of energy conservation
- P1 + ½v12 + gy1 = P2 + ½v22 + gy2
- Points 1 and 2 are two different locations in the “pipe” where the pressure might be different, the fluid speed may be different, the cross-sectional area may be different, and the height of the pipe might be different.
- We assume that the fluid is incompressible, so the density, , does not change. [Technically, we also assume the fluid is nonviscous, but we are not covering viscosity in Phy 122.]
[Skip 15.9: Viscosity and Surface Tension]
The Ideal Spring (Review)
Hooke’s Law (Restoring force of an ideal spring): F = -kx
- the minus sign means that the restoring force always points the opposite of the direction of the displacement.
- k is the spring constant (units: N/m)
- x is the displacement of the spring from its unstrained length
Simple Harmonic Motion and Circular Motion (Reference Circle) (13.1-13.3)
We use the concept of the reference circle to derive formulas to describe the displacement, velocity, and acceleration of an object undergoing simple harmonic motion.
Period (the time for one cycle): T = 1/f = 2π/ ω(units: s)
Frequency (number of cycles per second): f = 1/T(units: Hz = 1/s)
Angular frequency: ω = 2πf = 2π/T(units: rad/s = s-1)
(Do NOT confuse the angular frequency, ω, of an object in simple harmonic motion with the angular velocity, ω, of a body undergoing rotation or circular motion. We relate them when comparing the motion of an object in uniform circular motion and the projection of that motion on the screen/x-axis.)
Amplitude: the object’s maximum displacement from equilibrium. Object oscillates between x = -A and x = +A.
Displacement: x = A cos(ωt) = A cos(SI unit: m)
Velocity: v = -Aω sin(ωt)(SI unit: m/s)
-maximum velocity occurs at x = 0: vmax = Aω
-zero velocity occurs at x = +A and x = -A
Acceleration: a = -Aω2cos(ωt)(SI unit: m/s2)
- maximum acceleration occurs at x = +A and x = -A: amax = Aω2
Period/Frequency of vibration depends on mass and spring constant (13.4)
Angular frequency: orPeriod:
Vertical Spring (13.4)
If the ideal spring is vertical, the motion is still simple harmonic motion. The spring will oscillate about an equilibrium position, y0 below the unstrained position (the unstrained position is the spring without any mass hanging off it).
Equilibrium position:
Displacement: y = A cos(ωt)
(The rest as before for the horizontal spring, just using y instead of x.)
Energy and Simple Harmonic Motion (13.5)
Elastic Potential Energy: (SI units: J)
Total Mechanical Energy for a horizontal simple spring:
If there are no nonconservative forces (like friction), then mechanical energy is conserved: E0 = Ef.
For a more complex system/object, we can write out the generic form of the total mechanical energy formula to include elastic potential energy.
If you start out with this most generic form of mechanical energy, then you simply cross out which terms do not apply to your situation. For example, if there is nothing rotating, then cross out the rotational kinetic energy term.
Once again, if there are no nonconservative forces (like friction), then mechanical energy is conserved: E0 = Ef.
Simple Pendulum (13.6)
A simple pendulum is a mass, m, suspended at the end of a light string or rod of length, L.
For small angles only:
Angular frequency: (for small angles)
Period: (for small angles)
Notice that the period/frequency of a pendulum only depends on the length of the pendulum and the acceleration due to gravity, NOT the mass hung at the end.
Physical Pendulum (13.6)
Solid object that swings back and forth on a pivot under the influence of gravity
-I = moment of inertia of object
-l = distance between center of mass and axis of rotation
-g = acceleration due to gravity
-m = mass of physical pendulum
Damped Oscillations (13.7)
-Drag force on a slowly moving object: F = -bv
-Damping constant, b, depends on shape of object and viscosity of air or other medium
displacement of a damped oscillator
b = damping constant; Si units = kg/s
Underdamped
Amplitude of oscillations decays exponentially over time:
A0 = initial amplitude of oscillation
Critically damped– damping is increased such that the system doesn’t oscillate but merely relaxes back to equilibrium
Overdamped–damping is greater than critically damped; no oscillation; takes longer to relax back to equilibrium
Driven Oscillations and Resonance (13.8)
- simple example is pushing a child on a swing
- natural frequency, f0: frequency that system oscillates at if left to itself
- e.g. natural frequency of a pendulum = sqrt(g/l)/2π
- e.g. natural frequency of a spring = sqrt(k/m)/2π
- driving frequency, fext: frequency of periodic external force applied to oscillating system
- response (amplitude) of oscillation is larger the closer the driving frequency is to the natural frequency of the oscillating system
- largest response (amplitude) when fext = f0. Called resonance.
Waves and Sound (14.1)
Two features common to all waves:
1) they are a traveling disturbance
2) waves carry energy from place to place
Two basic types of waves are transverse and longitudinal waves.
For periodic waves, we can define maximum amplitude, A, period, T, frequency, f = 1/T, and wavelength, λ.
Wavelength, λ: distance over which a wave repeats itself(See also fig. 14.6)
Period, T: repeat time of wave
Frequency, f: inverse of period
Maximum amplitude, A: maximum motion of particles of the medium as the wave passes through
Speed of a wave: (true for any wave)
Speed of a Wave on a String (14.2)
Speed of a wave on a string: (this eqn only applies to a wave on a string)
- where FT is the tension force in the string, and ( = m/L) is the linear density (or mass per unit length) of the string
Reflections at end of string
- wave pulsed is inverted in reflection off of fixed (tied) end
- wave pulsed keeps original orientation in reflection off of end that is free to mov
Harmonic Wave Function (14.3)
We can describe waves mathematically using the following formulas:
Wave moves toward +x direction:
Wave moves toward -x direction:
- In the above two equations, t and x are variables (where you are in time and space), and A, f, and λ are set/defined for the particular wave being described by the formula. The variable y refers to the motion of the particles/medium through which the wave travels.
Speed of Sound (14.4)
- Sound is a longitudinal pressure wave in air (usually in air, although sound also travels thru liquids and solids)
- speed of sound in air (at room temperature, 20 C) = 343 m/s
- frequency determines pitch of sound
- all frequencies travel at the same speed in the same medium
- sound requires a medium (substance) to travel through. It does not travel through a vacuum (like space)
Sound Intensity (14.5)
Waves carry energy. Power = energy transported/unit time. (
P = E/tSI units = J/s = W = watt)
Intensity of sound determines how loud it is.
Sound Intensity:(SI units = W/m2)
- where P = power, and A = unit area the sound is distributed over.
- if the sound comes from a point source (power spread out equally in all directions) then
A = 4πr2 (the surface area of a sphere of radius r)
Intensity level in decibels: (units = dB = decibels)
- where “log” means logarithm to the base 10 (NOT the natural logarithm)
- I is the sound intensity in question
- I0 is the intensity of the reference level (usually taken to be the threshold of hearing for humans, 1x10-12 W/m2).
- Note: sound intensity, I (in W/m2) and intensity level, β (in dB) are not the same thing, although they are related to each other.
- intensity level is given in a logarithmic scale.
- Make sure you understand how to go from I to β, and β to I.
- Review Appendix A-5 in the textbook (Walker, 4th edition) if your math skills in exponents and logarithms are a bit “rusty”
Doppler Effect (14.6)
- the Doppler effect is the apparent change in frequency due to the motion of a sound source (and/or observer)
General case (both source and observer moving):
- where fo is the frequency “heard” by the observer, fs is the frequency emitted by the source, vo is the speed of the observer, vs is the speed of the source, and v is the speed of sound.
- Note: the speed of the wave (sound in this case) is determined by the properties of the medium (substance) that it propagates through, and doesn’t depend on the speed of the source.
- in the numerator, use the + sign if the observer is moving toward the source, and use the – sign if the observer is moving away from the source
- in the denominator, use the – sign if the source is moving toward the observer, and use the + sign if the source is moving away from the observer.
- if the observer is NOT moving, then vo = 0.
- if the source is NOT moving, then vs = 0.
Linear Superposition, Constructive and Destructive Interference of Sound Waves (14.7)
- superposition occurs when two waves are traveling thru the same space at the same time. The resultant disturbance is the sum of the two waves.
- constructive interference occurs when two waves are exactly in phase (crests line up)
- destructive interference occurs when two waves are exactly out of phase (the crests from one wave line up with the wave troughs from the second wave)
Note: waves don’t disappear or stop traveling through the medium if destructive interference occurs. Only at the location(s) of the destructive interference, do the waves cancel each other out. The rest of the medium where the waves travel, still have the wave(s).
- If you have two wave sources vibrating in phase, then you can get constructive or destructive interference at different locations depending on the difference in path length from the two sources, to the location in question.
Two sources in phase means they are both emitting wave crests at the same time.
Constructive interference: L2 – L1 = ΔL = nλ(sources in phase)
Destructive interference: L2 – L1 = ΔL = (n+½)λ(sources in phase)
- where λ is wavelength
- ΔL is the difference in path length
- path length is the straight line distance from source to observer
- n = 0, 1, 2, 3, …
Two sources with opposite phase, means that one source emits a wave crest at the exact same time that the other source emits a wave trough (you can do this by reversing the wires leading from amplifier to speaker, like the in class demonstration).
Destructive interference: L2 – L1 = ΔL = nλ(sources of opposite phase)
Constructive interference: L2 – L1 = ΔL = (n+½)λ(sources of opposite phase)
- where λ is wavelength, ΔL is the difference in path length, andn = 0, 1, 2, 3, …
Transverse and Longitudinal Standing Waves (14.8)
- another case of interference can occur when we have waves reflected back and forth – two waves traveling in the opposite direction, but over the same space = standing waves.
- a standing wave oscillates with time, but it’s location is fixed
- only certain conditions will produce certain standing waves patterns.
- for standing waves on a string, this condition is related to the speed of waves on a string.
Transverse standing waves on a string (string fixed at both ends):
(n = 1, 2, 3, …)
- n = 1 is the fundamental frequency, n = 2 is the 2nd harmonic (f2 = 2 f1), n = 3 is 3rd harmonic (f3 = 3 f1), etc
Longitudinal standing waves - resonance in air column (tube open/closed both ends):
(n = 1, 2, 3, …)
Longitudinal standing waves – resonance in an air column (tube open one end & closed at other end):
(nodd = 1, 3, 5, …)
Beats (14.9)
Beats “can be thought of as an interference pattern in time.” Pg 485, Physics, 4th editions, J.S. Walker, Addison-Wesley, 2010.
- beats occur when you add two waves with slightly different frequencies.
- as the waves interfere with each other, the intensity changes in intensity in a pattern
(it gets louder and softer in a repeating pattern)
Beat frequency: fbeat = |f1– f2|(SI units: 1/s)
A beat frequency of 4 Hz means the ear hears maximum loudness 4 times a second.
Beats are used to tune musical instruments so they produce exactly the same frequency sound for a particular musical note. Humans can only hear beat frequencies of 15-20 Hz, so the frequencies being tuned must be close to start with.
Reflection of Light (26.1)
Law of Reflection:θi = θr
- the angle of incidence and the angle of reflection are measured relative to the normal to the surface (normal to the surface means 90 degrees from the surface)
Plane Mirrors (26.2)
You see an image of an object
1) Image is upright
2) Image is the same size as the object (hi = ho)
3) Image is located same distance “behind” mirror as object is in “front” (di = do)
First example of a virtual image – the rays do not actually emerge from the virtual image, they only appear to do so
Spherical Mirrors (26.3-26.4)
Concave Mirror: focal lengthf = +½ R
Convex Mirror: focal lengthf = -½ R
Concave Mirror
Object Placement / Image Location/Type / Image Size / Image OrientationBetween C & F / Beyond C/Real / Enlarged / Inverted
Beyond C / Between C & F/Real / Reduced / Inverted
Between F and Mirror / Behind Mirror/Virtual / Enlarged / Upright
Convex Mirror
Object Placement / Image Location/Type / Image Size / Image OrientationAnywhere / Virtual / Reduced / Upright
Mirror Equation:
Magnification Equation:
Sign Conventions
- Real images on front side of mirror:di = +
- Virtual images behind mirror:di= -
- Image upright:m = +
- Image inverted:m = -
- Image enlarged:m > 1
- Image reduced:m < 1
Refraction of Light (26.5)
Index of Refraction:(n ≥ 1)
Wavelength of light in a material:
Because light travels at different velocities in different materials, light “bends” or refracts at the interface between two materials.
Snell’s Law:
- for light traveling from material 1 to material 2
- if n1n2, the light is bent “toward” the normal (θ2θ1)
- if n1n2, the light is bent “away” the normal (θ2θ1)
Apparent depth: depth an object appears to be
- objects in water often appear bent or shallower than they really are because of the bending of the light ray at the interface between the air and the water
Critical Angle:(n1n2)
- Total Internal Reflection: for angles at the critical angle and bigger, no light is transmitted into material 2; the light ray just skims along the interface between the two materials.
Lenses (26.6-26.7)
In this class, to minimize confusion, we always work from left to right. So a real object is to the left of the (first) lens. In real life, of course light can go left to right or right to left through a lens.
Converging Lens
Object Placement / Image Type / Image Size / Image Orientation / ExampleBeyond 2F / Real / Reduced / Inverted / Camera
Between 2F & F / Real / Enlarged / Inverted / Film projector
Between F and Lens / Virtual / Enlarged / Upright / Magnifying glass
Diverging Lens
Object Placement / Image Location/Type / Image Size / Image OrientationAnywhere / Virtual / Reduced / Upright
Thin Lens Equation:
Magnification Equation:
Sign Conventions
- Focal Length
converging lens:f = +
diverging lens:f = -
- Object Distance
object to left of the lens (real object):do = +
“object” (image from first lens) to right of 2nd lens (virtual object):do = -
- Image Distance
image formed to right of lens (real image):di = +
image formed to left of lens (virtual image):di = -
- Magnification
image is upright:m = +
image is inverted:m = -
Lenses in combination (27.2): with a multiple lens system, you apply the thin lens equation to each lens separately to find the location of the final image. The image from the first lens serves as the object for the second lens.
Dispersion and the Rainbow (26.8)
The index of refraction of a material depends slightly on the wavelength of light. This leads to dispersion – the spreading of light into its color components. Examples of dispersion of light: prisms and rainbows.
Human Eye and Corrective Optics (27.1-2)
Near point: the shortest distance that you can achieve a sharp focas
Lens has highest curvature (eye muscles tense)
About 25 cm for young people
Far point: the greatest distance an object can be and still be in focus
Lens relatively flat (eye muscles relaxed)
Really large (approximately infinity) for people with normal vision
F-number for cameras (skip Spring 2011)
f-number = focal length/diameter of aperature = f/D
Nearsighted: can see close clearly but not far (far point is NOT infinity)
Image focused in front of retina
Correct with a diverging lens
Farsighted: can see clearly far, but not up close (near point is much farther from eye than 25 cm)
Image focused “behind” the retina (if the light rays could go through the retima)
Correct with a converging lens
The Magnifying Glass (27.3)
Magnifying glass works by allowing things to be viewed closer than the near point so it appears larger
Angular Magnification: M = ’/ = N/f
(N = distance of near point from the eye; f = focal length of magnifying glass)