Download PHYS 1111 Exam 2 Formula Sheet

PHYS 1111 Exam 2 Formula Sheet
Displacement: ∆x = xf – xi
Average speed = distance/time
Average velocity: vav = ∆x/∆t
Average acceleration: aav = ∆v/∆t
1-dimensional motion with constant acceleration:
v = v0 + at
vav = 1/2 (v0 + v )
x = x0 + 1/2 (v0 + v ) t
x = x0 + v0 t + 1/2 at2
v2 = v02 + 2a(x – x0)
Acceleration due to gravity: g = 9.81 m/s2
Components of a vector:
x component of vector A: Ax = A cos θ, where θ is measured relative to the x-axis.
y component of vector A: Ay = A sin θ, where θ is measured relative to the x-axis.
Magnitude of vector A:
A = √(Ax2 + Ay2 )
Direction angle of vector A: θ = tan-1 (Ay / Ax )
Unit vectors, vector addition:
A + B = (Ax + Bx) x + (Ay + By) y
Relative motion:
v13 = v12 + v23
Projectile motion:
ax = 0
ay = – g
General launch angle from origin (x0 = y0 = 0) with initial speed v0 at an angle θ
with respect to the horizontal:
x = (v0 cos θ) t
y = (v0 sin θ) t – 1/2 gt2
vx = v0 cos θ
vy = v0 sin θ – gt
vx 2 = v02 cos2 θ
vy 2 = v02 sin2 θ – 2 g ∆y
Range:
R = (v02/g) sin 2θ
Maximum Height:
ymax = (v02 sin2 θ)/2g
Other info:
Volume = Area × depth = 4πR3/3
1 in = 0.0254 m
1 µm = 10−6 m
Newton’s Second Law of Motion (component form):
ax = ΣFx/m
ay = ΣFy/m
az = ΣFz/m (1 N = 1 kg m/s2)
Weight:
W = mg
Kinetic friction:
fk = µk N
Hooke’s Law:
Fx = kx
Uniform circular motion:
W = Fd cos θ (1 J = 1 N m = 1 kg m2/s2)
K = 1/2 mv2
Work-Energy Theorem:
Wtotal = ΔK = 1/2 mvf2 – 1/2 mvi2
Work done by a spring:
W = 1/2 kx2
Power:
fs,max = µs N
fcp = macp = mv2/r
Work done by a constant force:
Kinetic Energy:
Static friction:
P = W/t = Fv (1 W = 1 J/s = 1 kg m2/s3; 1hp = 746W)
Work done by a conservative force: Wc = – ΔU = Ui – Uf
Gravitational potential energy:
Spring potential energy:
U = mgy
U = 1/2 kx2
Conservation of mechanical energy: E = U + K
Work done by a nonconservative force:
Linear momentum:
p = mv
Newton’s Second Law:
Wnc = ΔE = Ef – Ei
p total = p 1 + p 2 + p 3 + …
Σ F = Δp/Δt
Impulse:
I = FavΔt = Δp
Inelastic collision in one dimension: vf = (m1 v1,i+ m2 v2,i) / (m1 + m2)
Elastic collision in one dimension:
v1,f = vo (m1 – m2 ) / (m1 + m2)
v2,f = 2 m1 vo / (m1 + m2)
Location of center of mass:
Xcm = (Σ mx)/M
Ycm = (Σ my)/M
Motion of center of mass:
Vcm = (Σ mv)/M
Acm = (Σ ma)/M
MAcm = Fnet,ext
Angular position:
θ (in radians) = s / r
Average angular velocity:
ωav = Δθ/Δt
Average angular acceleration:
αav = Δω/Δt
Period of rotation:
T = 2π/ω
Rotational kinematic equations:
ω = ωo + αt
θ = θo + 1/2 (ωo + ω) t
θ = θo + ωo t + 1/2 α t2
ω2 = ωo2 + 2 α (θ - θo)
Tangential speed:
vt = rω
Centripetal acceleration:
acp = rω2
Tangential acceleration:
at = rα
Rolling motion:
ω=v/r
Kinetic energy of a rotating object: K = 1/2 I ω2
Moment of inertia:
I = Σ mi ri2
Kinetic energy of rolling motion:
K = 1/2 mv2+ 1/2 I ω2 = 1/2 mv2 (1 + I / m r2)
Torque for a general force:
τ = r F sin θ
Newton's second law for rotation:
τ = Iα = ΔL/Δt
Conditions for static equilibrium:
Σ Fx = Σ Fy = Σ τ = 0
Angular momentum:
L = Iω = r m v sin θ
Work done by a torque:
W = τ Δθ
Newton's Law of Universal Gravitation:
F = G m1 m2 / r2
Acceleration of gravity:
g = G ME/ RE2
Kepler's third law:
T = (2π/√(GM)) r3/2
Gravitational potential energy:
U = – G m1 m2 / r
Total mechanical energy:
E = K + U = 1/2 mv2 – G m1 m2 / r
Escape speed:
ve = √(2GM/R)
Frequency:
f = 1/T
Angular frequency:
ω = 2πf = 2π/T
Position versus time in SHM:
x = A cos (2πt/T) = A cos (ωt)
Velocity as a function of time in SHM:
v = – Aω sin (ωt)
Acceleration as a function of time in SHM:
a = – Aω2 cos (ωt)
Maximum speed of an object in SHM:
vmax = Aω
Maximum acceleration of an object in SHM:
amax = Aω2
Period of a mass on a spring in SHM:
T = 2π √(m/k)
Total energy of a mass on a spring in SHM:
E = 1/2 kA2
Potential energy of a mass on a spring in SHM:
U = 1/2 kA2 cos2 (ωt)
Kinetic energy of a mass on a spring in SHM:
K = 1/2 kA2 sin2 (ωt)
Period of a simple pendulum of length L:
T = 2π √(L/g)
Relation of wavelength, frequency, and speed:
v = λf
Mass per unit length of a string:
µ = m/L
Speed of a wave on a string:
v = √(F/µ)
Sound intensity:
I = P/A
Sound intensity for a point source:
I = P/4πr2
Intensity level in decibels:
β = 10 log (I/Io)
Doppler effect – moving observer:
f ′ = (1 ± u/v)f
Doppler effect – moving source:
f ′ = (1/(1 m u/v))f
Doppler effect – general case:
f ′ = (1 ± uo/v) /(1 m us/v) f
Superposition:
ytotal = y1 + y2
Standing waves on a string:
fn = n f1 = n(v/2L)
λn = λ1/n = 2L/n
Standing waves in vibrating columns of air:
Closed at one end:
fn = n f1 = n(v/4L)
λn = λ1/n = 4L/n
Open at both ends:
Exponents and logarithms:
fn = n f1 = n(v/2L)
λn = λ1/n = 2L/n
for n = 1, 3, 5, …
for n = 1, 2, 3, …
x = 10n
n = log x
log (xy) = log x + log y
log (x/y) = log x – log y
Temperature conversion relations:
TF = 9 TC/5 + 32
TC = 5/9 (TF – 32)
T = TC + 273.15
Thermal expansion:
linear: ΔL = α Lo ΔT
area: ΔA = 2α Ao ΔT
volume: ΔV = 3α Vo ΔT
Heat capacity:
C = Q/ΔT
Specific heat:
c = Q/(m ΔT )
Conductive heat flow:
Q/t = kA (ΔT/L)
Equation of state for an ideal gas:
PV = NkT = nRT
Molecular mass:
M = NA m
Boyle’s Law:
PV = constant
Charles’s Law:
V/T = constant
Kinetic energy and temperature:
(1/2 mv2)av = Kav = 3/2 kT
rms speed:
vrms = √(3kT/m) = √(3RT/M)
Internal energy of an ideal gas:
ΔU = 3/2 NkT = 3/2 nRT
Definition of latent heat:
Q = mL
First Law of Thermodynamics:
ΔU = Q – W
Work done at constant pressure:
W = P ΔV
Work done in isothermal process:
W = nRT ln (Vf/Vi)
Molar specific heat:
Q = nCΔT
where CV = 3/2 R and CP = 5/2 R
Adiabatic process:
PVγ = constant
where γ = CP/CV
Heat engine efficiency:
e = W/Qh = (Qh – Qc)/Qh = 1 – (Qc/Qh)
Maximum efficiency:
emax = 1 – (Tc/Th)
Maximum work:
Wmax = emax Qh = 1 – (Tc/Th) Qh
Coefficient of Performance:
refrigerator or A/C:
heat pump:
COP = Qc/W
COP = Qh/W
Change in entropy:
ΔS = Q/T