Vectors uations of Motion
5
Phys 1301 Final Exam Formula Sheet t v a t x v s m . g Δt Δv a Δt Δx v v v v v v y B A x B A B A A A y A x A A t t avg avg y y x x y x y x Δ Δ = Δ Δ = = = = = − = + = + + + = + + = + = → Δ → Δ → → → → → → → → 0 0 2 21 12 23 12 13 2 / 1 2 2 lim lim time distance speed 81 9 ˆ ) ( ˆ ) ( ) ( A ˆ ˆ Equations General motion Relative ition Vector Add Vectors t v v y y t v v x x t a t v y y t a t v x x y y a v v x x a v v t a v v t a v v y y x x y y x x y y y x x x y y y x x x ) ( 2 1 ) ( 2 1 2 1 2 1 ) ( 2 ) ( 2 0 0 0 0 2 0 0 2 0 0 0 2 0 2 0 2 0 2 0 0 + + = + + = + + = + + = − + = − + = + = + = Motion of uations q E Projectile motion: make a x =0, a y = - 9.81m/s 2 θ ⋅ = θ ⋅ = sin cos 0 0 0 0 v v v v y x centripetal force: F c = mv 2 /r equilibrium conditions ΣF x =0, ΣF y =0 non-equilibrium conditions ΣF x =ma x , ΣF y =ma y static friction force f s ≤μ s N kinetic friction force f k =μ k N spring force: F s =-kx
Transcript of Vectors uations of Motion
Phys 1301 Final Exam Formula Sheet
tva
txv
sm.g
ΔtΔva
ΔtΔxv
vv
vvv
yBAxBABA
AA
yAxAA
t
t
avg
avg
yyxx
yx
yx
ΔΔ
=
ΔΔ
=
=
=
=
=
−=
+=
+++=+
+=
+=
→Δ
→Δ
→→
→→→
→→
→
0
0
2
2112
231213
2/122
lim
lim
timedistancespeed
819
ˆ)(ˆ)(
)(A
ˆˆ
Equations General
motion Relative
itionVector Add
Vectors
tvvyy
tvvxx
tatvyy
tatvxx
yyavv
xxavv
tavvtavv
yy
xx
yy
xx
yyy
xxx
yyy
xxx
)(21
)(21
2121
)(2
)(2
00
00
200
200
020
20
20
2
0
0
++=
++=
++=
++=
−+=
−+=
+=
+=
Motion of uationsqE
Projectile motion: make ax=0, ay= - 9.81m/s2
θ⋅=
θ⋅=
sincos
00
00
vvvv
y
x
centripetal force: Fc= mv2/r equilibrium conditions à ΣFx=0, ΣFy=0
non-equilibrium conditions à ΣFx=max, ΣFy=may static friction force fs≤μsN kinetic friction force fk=μkN spring force: Fs=-kx
€
Work and Energy equationsW = (F cosθ)dWTotal =W1 +W2 +W3 + ...WTotal = (FTotal cosθ)dWTotal = ΔK
K =12mv 2
Wspring =12kx 2
P =Wt
P = Fv cosθ
Momentum .vmp
=
Conservation of momentum: m1v1i+m2v2i= m1v1f+m2v2f
Inelastic collision:
of vmmmmv ⎟⎟
⎠
⎞⎜⎜⎝
⎛
+
−=
21
211 (for a 1-dimensional elastic collision)
of vmmmv ⎟⎟
⎠
⎞⎜⎜⎝
⎛
+=
21
12
2 (for a 1-dimensional elastic collision)
Mmx
XCM∑=
Mm
CM∑= v
V
Mm
CM∑= a
A
€
Wc = −ΔU
€
Ugravity = mgyi 2
21 kxUspring =
€
E =U +K
€
Wnc = ΔE = E f − Ei
Rotational Kinematics and Energy Average angular velocity: ω θ
av =Δ
Δt Tangential speed: v! = rω
Instantaneous angular velocity: ω
θ=
→limΔ
Δ
Δt t0
Period of rotation: T = 2π
ω
Average angular acceleration: α
ωav =
Δ
Δt
Instantaneous angular acceleration: α ω=
→limΔ
Δ
Δt t0
Angular kinematic equations:
ω ω α= +0 t
θ θ ω ω= + +012 0( )t
θ θ ω α= + +0 012
2t t
ω ω α θ θ202
02= + −( )
Centripetal acceleration: a rcp = ω 2=v2/r
Tangential acceleration: a rt = α
Rotational kinetic energy: K I= 12
2ω
Moment of inertia: I m r=∑ i i2
Kinetic energy of rolling motion: K =!!mv! + !
!Iω!
Mo
Moment of inertia of objects: 𝐼!"#$ =!!𝑀𝑅!
𝐼!"# =!!"𝑀𝐿! axis through midpoint
𝐼!"# =!!𝑀𝐿! axis at one end
Rotational Dynamics and Static Equilibrium Torque (Tangential force): τ = rF
Torque (general definition):τ θ= rF sin
Newton’s law for rotational motion: τ α∑ = =I L
tΔ
Δ
Angular momentum: L I= ω
Equilibrium conditions: ΣFx=0; ΣFy=0; Στ=0
Work done by torque: W = τ θΔ
Work-energy theorem: W K K K= = −Δ f i
Angular momentum for point particle: L rmv= sinθ
Gravity
Gravitational force between m1 and m2: 221
rmmGF =
Gravitational acceleration on the surface of the Earth: 2E
E
RGM
g =
Gravitational acceleration on the surface of a planet P: g = GMP
RP2
Period of rotation around the sun: T = 2π
GMS
!
"##
$
%&&(r)
32
Gravitational potential energy: rmmGU 21−=
Escape speed for the Earth: E
Ee R
GMv
2=
Mechanical energy: UKE +=
2211 /1067.6 kgNmxG −= kgxM Earth
241097.5= kgxM sun
301000.2= kmxREarth31037.6=
Oscillations
Frequency of oscillation: Tf 1=
Position versus time in Simple Harmonic Motion: ( )tAx ωcos=
Velocity in Simple Harmonic Motion: v =- Aωsin(ωt) ; vmax=Aω
Acceleration in Simple Harmonic Motion: a =- Aω2cos(ωt); amax= Aω2
Angular frequency of a mass on a spring: mk
=ω
Period of a mass on a spring: kmT π2=
Tf ππω
22 ==
Potential energy in oscillatory motion: )(cos
21 22 tkAU ω=
Kinetic energy in oscillatory motion: )(sin
21 22 tkAK ω=
Total energy in oscillatory motion: 2
21 kAE =
Period of a Pendulum: gLT π2=
Sound and Waves
speed of a wave: v=λf
speed of a wave on a string:
harmonics for string or pipe with both end open: , n=1,2,3…
harmonics in a pipe, one end open: , n=1,3,5…
Doppler effect: f ' = f (1± vo / v1 vs / v
)
Constructive interference: d2-d1=nλ, n=1,2,3…
Destructive interference: d2-d1=(2n+1) λ2
, n=0,1,2,3…
