Download Physics 51 "Study Guide" for Final Exam ("Laundry List" of important

Physics 50
"Study Guide" for Final
("Laundry List" of important concepts)
Concept
(important concepts in bold; vectors also shown in bold) Symbol or Equation
We use the international SI system of units which employs the meter (m)
for length, the kilogram (kg) for mass, and the second (s) for time
Trigonometry: We use sine, cosine, and tangent functions to relate
angles to the ratios of various lengths of the sides of right triangles
Angles can be measured in radians (2 π rads per circle) or in degrees
(360˚ / circle). Set your calculator correctly.
Physics quantities are either scalars or vectors (magnitude & direction)
Unit vectors give only directional information, eg. î for x-axis, ĵ for y
When adding vectors, think of the vectors' components separately
Scalar product ("dot product") of two vectors gives a scalar result
Vector product ("cross product") of two vectors gives a vector result
The direction of the cross product vector is perpendicular to both A & B;
The position vector r of a point P is the vector from the origin to P
by Todd Sauke
kilo (k)=103 mega (M)=106
milli (m)=10-3 micro (μ)=10-6
sin(θ) = ho/h θ = sin-1(ho/h)
cos(θ) = ha/h θ = cos-1(ha/h)
tan(θ) = ho/ha θ = tan-1(ho/ha)
h2 = ho2 + ha2 (Pythagoras)
|v| = a positive scalar, "v"
(scalar) * (unit vector) = vector
components of vectors add
A • B = A B cos (φ)
|A x B| = A B sin(φ)
given by right-hand-rule (RHR)
r =xî +yĵ +zǩ
Kinematics describes motion. Dynamics deals with effect of forces on motion Δr = r – r0
Δt = t – t0
Average velocity: vave=Δr/Δt . Instantaneous velocity: v=lim(ΔtÆ0) Δr/Δt . speed (scalar); velocity (vector)
Average acceleration: aave=Δv/Δt . Instantaneous acceleration: a=lim(ΔtÆ0) Δv/Δt . time (scalar);acceleration (vector)
Speed is different from velocity. Speed is the (positive) magnitude of velocity
We use 6 kinematic variables: [x (or y), x0 (or y0), vx (or vy), vx0 (or vy0) , ax (or ay) and t] label directions '+' & '-' on diagram
Constant acceleration: make a table of known (or implied) kinematic variables v = v0 + a t
x-x0 = ½(v0+v) t
2
2
and select the equation here involving those variables and the target variable. v = v0 + 2a (x – x0)
In two (or three) dimensions, apply the x, y (or z) equations separately
Projectile motion: an object flies freely through the air, accelerated only by gravity
The Quadratic Formula comes in handy: for
a x2 + b x + c = 0
Æ
x = x0 + v0 t + ½ at2
ay= -g = -9.8 m/s2 (projectile motion)
x = (-b ± √ b2 – 4ac ) / (2a)
Velocity of object A relative to object B is written vAB (note order of subscripts) vAC = vAB + vBC
Acceleration component parallel to v changes the speed of the object
Acceleration component perpendicular to v only changes its direction
vBA = -vAB
Force is a (vector) measure of interaction between bodies. Resultant force = vector sum R = F1 + F2 + F3 + . . . = Σ F
Newton's 1st Law of motion: An object continues in a state of rest (or motion "Inertia" is what does it via
at constant speed along straight line) unless compelled to change by net force
mass (kg) mass & weight not same
nd
Newton's 2 Law of motion: Net external force gives proportional acceleration Σ Fext = m a (SI newton, "N")
Newton's 3rd Law of motion: When 'A' pushes 'B', 'B' pushes 'A' back
FAB = -FBA ("action-reaction")
Different types of forces all add together to make the "net force" in #2
w = mg
weight is a force
Free body diagrams help solve problems with force and acceleration.
The free body diagram shows a given body "free" from its surroundings.
The diagram shows labeled vectors of magnitude and direction for all
forces applied to the body by other bodies. Don't include bogus forces!
Label axis directions! With all forces included, add components & apply
Newton's 2nd Law for acceleration of the body along the axis directions.
Mass and weight are not the same; weight is the gravity force on mass
The "normal force" is a force of constraint preventing material overlap
Friction forces oppose motion. The constraint of static friction has a limit.
Kinetic (sliding) friction is proportional to FN, and is speed independent
Tension force T is a pulling force transmitted along a rope, or wire
Tension is also a force of constraint, determined by other considerations
At equilibrium (at rest or at constant velocity) net force on body must be 0
Uniform circular motion can be specified by the period T of the motion
Uniform circular motion has constant speed, but not constant velocity!
The centripetal ("toward center") acceleration requires centripetal force
T
f
n
y
x
mg
Apparent weight = mg + ma
FN is ┴ to the surface
fsMAX = μs FN (parallel to surface)
fk = μk FN (note that μs > μk)
T at opposite ends acts oppositely
Use F=ma equations to get tension
ΣFext= 0
(use free body diagram)
T=2πr/v (v = tangential speed)
ac = v2/ r (centripetal acceleration)
Fc = m v2 / r (toward the center)
Work, a scalar with units of energy (SI joules, "J"), is the product of
W = F • s = F s cos(φ)
force acting through a distance. φ = angle between F and displacement s
Hooke's law for a spring says that the force required to compress
W = ∫ dW = ∫ F(x,y,z) • ds
F=kx
Fspring = - k x
(or stretch) a spring is proportional to the compression (or stretch)
Welastic=∫-k x dx = ½ kx12 – ½ kx22
The reaction force of the spring Fspring pushing back is equal and opposite
The kinetic energy K (a scalar) of a particle of mass m is the amount of work
K = ½ m v2
required to accelerate it from rest to speed v. The Work-Energy Theorem is: Wtotal = K2 – K1 = ΔK
Power P (a scalar, SI watt, W = J/s) is the time rate of doing work
Pav=ΔW/Δt
P=dW/dt
For a force F acting on a particle with velocity v, P is the scalar product: P = F • v
All forces are either conservative or nonconservative. For a conservative Wgrav = mgy1 –mgy2 =U1–U2 = -ΔU
force, the work-energy relation is completely reversible, mechanical energy
is conserved, and the work can be represented by a potential energy function
The sum of kinetic and potential energy is Total Mechanical Energy
If only conservative forces are acting, the Work-Energy relation yields:
When "other" (nonconservative) forces do work (Wother) we have:
The work done by nonconservative forces changes the internal energy
The sum of kinetic, potential, and internal energy is always conserved
For one dimensional (straight line) motion, force is related to U as:
Wel =½ kx12 – ½ kx22=U1–U2 = -ΔU
U is a function of position alone
Mechanical Energy = K + U
K1 + U1 = K2 + U2
K1 + U1 + Wother = K2 + U2
ΔK + ΔU +ΔUint = 0
Fx(x) = - dU(x) / dx
∂U
∂U
∂U
In three dimensions we use the sum of partial derivatives: F = -Grad(U)= -Î(U) F = - ( /∂x î + /∂y ĵ + /∂z ǩ)
The momentum p of a particle of mass m is a vector quantity given by: p = m v
v = particle velocity
nd
Newton's 2 lawÆ net force on a particle = time rate of change of p
ΣF = dp/dt
Impulse J, a vector, is the product of force acting on a particle over time J = F (t2 – t1)
J = ∫ F(t) dt
The Impulse-Momentum Theorem is obtained by integrating F=dp/dt Jtotal = p2 – p1 = Δp
The momentum of a particle is the impulse that accelerated it from rest
Total momentum of a system of particles is the vector sum of the individual momenta
P =pA+pB+ • • • = mAvA + mBvB+ • • •
B
B
B
If the net external force on a system is zero, the total momentum of the
If ΣFext = 0 Æ P = constant
system is constant (conserved): "Conservation of Momentum"
In collisions of all kinds, (elastic and inelastic), momentum is conserved Pi = Pf (assuming no external forces)
In an elastic collision, the kinetic energy is also conserved
Ki = Kf additionally for elastic
In an elastic collision of two bodies, the magnitude of the relative velocity is unchanged Å |vai – vbi| = |vaf – vbf|
The above fact allows us to simplify many problems that would otherwise be difficult!
1-Dimensional elastic problems are especially easier to do if you apply the above fact
In 1-D, (vai – vbi) = –(vaf – vbf)
Center of mass, rcm, of a system of particles is given by a weighted average position rcm = (m1r1+ m2r2+ • •)/(m1+m2+ • •)
The center of mass is a kind of "stand-in" for the entire mass of the system P = m1v1+ m2v2+ • • • = M vcm
The center of mass moves as if the entire mass were concentrated there
Rotational kinematics describes rotational motion about an axis of rotation
θ is the angle of rotation about the axis; ω and α are its time derivatives
Each rotational kinematic variable is analogous to a translational one
The equations of rotational kinematics are exactly analogous to those for
translation; use the one that has the given info and the target variable
Each rotational kinematic variable is related to its corresponding
translational one by the factor r, (distance from the rotation axis)
With circular motion, centripetal acceleration can be expressed in ω
If rotational speed changes (nonuniform circular motion) atotal is sum:
Rolling without slipping: translational and rotational speeds are related
Rotational dynamics deals with effects of torque τ on rotational motion
Torque is a vector, the vector ("cross") product of position vector r & F
Torque 'tries' to produce rotation. At equilibrium, no change in motion:
ΣFext = M acm
(M = Σmi)
use radians for θ, ω, α !
ω= dθ/dt α= dω/dt = d2θ/dt2
θ:x, ω:v, α:a, time is the same
ω= ω0+αt θ - θ0= ½(ω0+ω) t
θ=θ0+ω0t+½αt2 ω2=ω02+2α(θ−θ0)
s=rθ
vT = r ω
aT = r α
'T' Æ 'tangential', s is along the arc
ac = vT2 / r Æ ac = r ω2
atotal = ac + aT
vcenter = R ω (R=rolling radius)
τ = F l (l = lever arm)
τ = r x F (direction from RHR)
ΣFext= 0 , Στext = 0
The center of gravity of a rigid body is the point at which its weight is
considered to act when the torque due to the weight is evaluated
Newton's 2nd Law for rotation is analogous to the linear (ΣF = ma)
τ is analogous to F; moment of inertia (I) to inertia (mass); α to a
The moment of inertia Icm of a body of mass M about an axis through the
center of mass is related to the Ip about a parallel axis at a distance d away
Some moments of inertia for some symmetrical objects: Hoop; I = M R2
Solid cylinder or disk; I = ½ M R2
Rod, ┴ axis thru center; I = 1/12 M L2
xcg = Σ Wi xi / Σ Wi
(similar to center of mass)
Σ τext = I α
(I = Σ m r2)
I depends on distribution of m
Ip = Icm + M d2
↑ Parallel axis theorem
2
Sphere; I = /5 M R2
Thin spherical shell; I = 2/3 M R2
∫ τ(θ) dθ
Rotational work and kinetic energy are also analogous to the linear kind
(just exchange translational and rotational analogous variables)
Angular momentum L is also analogous to the linear kind (p = m v)
WRot = τ Δθ
The angular momentum L of a particle with respect to a point O is related to
L = r x p = r x mv (L=vector)
its position vector r relative to O and its linear momentum p = m v
WRot =
P = τzωz
2
KRot = ½ I ω
L=Iω
(rigid body about symmetry axis)
(L is conserved when Στext=0)
Ktotal = ½ M vcm2 + ½ Icm ω2
Net external torque is equal to the time rate of change of angular momentum
Σ τext = dL/dt if τ=0ÆL=const
For a rigid body to be in equilibrium (no linear or angular acceleration): Σ F = 0, Σ τ = 0 (about any point)
The torque due to the weight of a body is as if the weight is applied at the rcg = rcm (for a uniform g field)
For a rigid body with both rotational and translational motion:
center of gravity, which is the same as the center of mass in a uniform g field. rcm = (m1r1+ m2r2+ • •)/(m1+m2+ • •)
Hooke's law: stress (force/unit area) is proportional to strain (fractional deformation) Stress / Strain = Elastic modulus
Tensile (stretching) and compressive deformation ÆYoung's modulus, Y
Y = tensile stress / tensile strain
Tensile stress = perpendicular force on area A, squeezing or stretching the body Y = (F┴ / A)/(Δl / l0)
Bulk strain (fractional volume change) results from bulk stress (pressure change) Bulk modulus, B = Δp/(ΔV/V0)
Shear strain (fractional sideways displacement) results from shear stress, F||/A
Shear modulus, S = (F||/A)/(x/h)
Most materials approximately obey Hooke's law up to a proportional limit. > elastic limit, things bend for good
Newton's law of gravitation: two bodies, masses m1, m2, distance r apart, attract Fg = G m1 m2 / r2
G, the universal gravitation constant is the same for any two masses
G = 6.67 x 10-11 N • m2 / kg2
Weight is the gravitational force on an object of mass m. (eg: @earth's surface) w = Fg = G ME m / RE2 = g m
The gravitational potential energy U of two masses is never positive:
U = - G m1 m2 / r
For circular orbits, centripetal acceleration comes from gravitational force v = √GME / r , period T = 2πr / v
Kepler's laws for planetary orbits: 1) ellipse w/ sun at focus 2) = area in = time 3) T = c a3/2, a = semi-major axis
Spherical mass distribution acts like its mass is all at its center for force on outside mass. Mass inside spherical shell, force=0
Periodic motion repeats in a definite cycle. Period T is the time for one cycle.
Frequency is measured in cycles per second (SI Hertz, "Hz"); ω = radians/s
When the net force on an object is a restoring force, directly proportional to
frequency, f = 1 / T = ω / 2π
the displacement x, the resulting motion is simple harmonic motion (SHM)
In SHM, displacement, velocity, and acceleration are sinusoidal in time.
Energy is conserved in SHM, oscillating between kinetic and potential
x = Acos(ωt+φ) ω = √ k/m
Fx = -kx
ax = Fx/m = -(k/m)x
A, φ come from initial conditions
constant E= ½mv2+½kx2 = ½kA2
2πf = ω = √ κ/I
(κ = -τz / θ)
A simple pendulum of length L executes SHM (for small excursions)
ω = √ g/L (independent of m!)
A physical pendulum is a rigid body suspended from an axis of rotation ω=√ mgd/I (for axis d from cg)
-(b/2m) t
If damping force F=-bv proportional to velocity is present Æ damped oscillator x = A e
cos(ω't)
Angular SHM: ω and f come from moment of inertia I and torsion constant κ
-
-
2
2
For small b, the system is underdamped, for b = 2√km it's critically damped ω' = √k/m – b /4m
When a sinusoidally varying driving force at ωd is applied to a damped harmonic A = Fmax / √ (k - mωd2)2+b2ωd2
oscillator, we have a forced oscillation at the applied frequency, w/ amplitude A for ωd near √k/m Æ resonance
Density ρ is mass per unit volume. Specific gravity is the ratio of ρ/ρwater ρ = m/V ρwater = 1000 kg/m3
Pressure p is normal force (F┴) per unit area. p = dF┴ /dA (SI Pascal, "Pa") pAtmospheric = 1.013 x 105 Pa
Pascal's law: pressure is transmitted undiminished throughout enclosed fluid gauge pressure = p - pAtmospheric
Pressure increases with depth below a static fluid's surface.
If ρ=constant Æ
p2–p1 = -ρg(y2 – y1); p=p0+ρgh
A fluid exerts an upward buoyant force on an immersed object (Archimedes)
Newton's third law means the immersed object exerts a reaction force on the fluid
An "ideal fluid" is incompressible and has no viscosity (no internal friction)
Fbuoyant = weight of displaced fluid
Ffluid (from emersed object) = - Fbuoyant
incompressible fluid: ρ=constant
The volume flow rate of a flowing fluid is area A times v (A ┴ to v)
dV/dt = Av
v = flow speed
The continuity equation expresses conservation of mass for a flowing fluid
ρ1A1v1 = ρ2A2v2
Bernoulli's equation relates pressure, flow speed, and elevation for any 2 points p1+ρgy1+½ρv12=p2+ρgy2+½ρv22
A wave is a disturbance that propagates from one region to another
A mechanical wave travels within some material medium with speed v v depends on properties of medium
A periodic wave repeats itself. The repeat time is T=1/ f. The repeat length is λ. v = λ f wave amplitude = A
A sinusoidal periodic wave has each point in SHM. For wave Æ +x axis y(x,t) = A cos(kx-ωt) (k=2π/λ)
The wave function y(x,t) describes the displacement of individual particles
The wave function obeys a partial differential equation, the wave equation
For wave Æ -x axis use "+" above
∂2y(x,t)/ ∂x2 = (1/v2) ∂2y(x,t)/ ∂t2
-
Speed of transverse waves on a string depends on tension F and mass density μ v = √F/μ
2
2
Wave motion transmits energy from one region to another. Average power Æ Pav=½√μF ω A (for sinusoid)
For waves that spread out in 3 dimensions, wave intensity I goes down w/ r2 I1/I2 = r22/r12 (I=Pav/unit area)
Total wave displacement where 2 waves overlap = sum of the individual displacements
A wave that reaches a boundary of a medium reflects back.
A wave traveling along a string with a fixed endpoint reflects an inverted wave
Sinusoidal incident and reflected waves combine to make a standing wave
Å principle of superposition
reflected wave superposes with 1st
two waves add up to 0 at endpoint
y(x,t) = ASW sin(kx) sin(ωt)
If both ends of a string of length L are fixed, standing waves must have L = n λ / 2
n = 1,2,3,…
an integer number of half wavelengths to "fit" and match end conditions fn = n v/2L
f1 = 1 / 2L √F/μ
-
Multiples of the n=1 fundamental frequency f1 are "harmonics" or "overtones"
n=2 Æ 2nd harmonic; first overtone
A sinusoidal sound wave is characterized by ω, f, λ, k and A, as above
A is the maximum displacement
pmax = BkA (k = wave number)
Sound is a longitudinal wave (motion of vibrating particles is parallel to wave motion) Sound is a pressure wave
The pressure amplitude pmax is the product of k, A and the bulk modulus B
The perceived loudness of a sound depends on its amplitude and frequency
The perceived pitch of a sound is primarily dependent on its frequency
The timbre (or "tone color") of a sound is determined by harmonic content
20 Hz < ~audible sound < 20,000 Hz
higher frequency Æ higher pitch
noise = all frequencies combined
The speed of sound v in a medium depends on the properties of the medium
v = √B/ρ (fluid); v = √Y/ρ (solid rod)
v = √γRT/M (gas) M=molar mass
R is the gas constant = 8.314 J/mol K. vsound ~ average speed of air molecule vsound in air ~344 m/s ~1130 ft/s
Sound speed in ideal gas depends on the ratio of heat capacities γ & temperature T
Sound waves transfer energy; the wave intensity I is Watts per meter2
The human ear is sensitive to sound over a broad range of intensities
Sound intensity level β using decibels (dB) is a logarithmic intensity scale
I= ½ √ρ B ω2A2 = ½ pmax2/√ρB
I0 = 10-12 W/m2 ~hearing threshold
β = (10 dB) log(I/I0)
When traveling sound waves reflect and interfere, standing sound waves are formed wave amplitude = 0 at a node
A pressure node is a place where pressure and density do not change
pressure node = displacement antinode
A displacement node is a place where molecular position does not change pressure antinode = displacement node
Sound in a gas: a solid boundary is a displacement node (pressure antinode) Boundary conditions determine
Open end of a pipe with sound is a pressure node (displacement antinode)
allowed λs for sound in pipes
An open pipe is an organ pipe that is open at both ends
fn = nv/2L (n=1,2,3,… open pipe)
A stopped pipe is open at the driven end, but closed at the other end
fn = nv/4L (n=1,3,5,…stopped pipe)
A resonant condition occurs for certain periodic driving force frequencies fdrive ~ fnormal mode Æ resonance
At resonance, the amplitude of the wave motion is maximum
Two or more sound waves overlapping in the same space Æ interference in phase Æ constructive interference
Wave interference can cause the canceling out of wave amplitude
out of phase Æ destructive interferance
Beats are heard when two tones with slightly different frequencies are sounded fbeat = fa – fb (beat frequency)
The Doppler effect for sound is the frequency shift that occurs when there fL = fS (v+vL)/(v+vS) (xS>xL)
is motion of a source of sound, a listener, or both, relative to the medium. L = listener, S = source, v=|vsound|
A sound source moving with speed vS greater than the speed of sound v
creates a shock wave with a wave front being a cone of angle α
sin(α) = v / vS
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version 1-12-2009(10)