2
Trigonometry and Vectors: sin 30 = cos 60 = 1 2 sin 36.9 cos 53.1 3 5 sin 45 = cos 45 = 1 2 sin 53.1 cos 36.9 4 5 sin 60 = cos 30 = 3 2 h adj = h cos θ = h sin φ h opp = h sin θ = h cos φ h 2 = h 2 adj + h 2 opp tan θ = hopp h adj h h opp φ θ h adj Law of cosines: C 2 = A 2 + B 2 2AB cos γ Law of sines: sin α A = sin β B = sin γ C B γ α C A β A = A x ˆ i + A y ˆ j + A z ˆ k ˆ A = A | A| A · B = A x B x + A y B y + A z B z = AB cos θ = A B = AB A × B =(A y B z A z B y ) ˆ i +(A z B x A x B z ) ˆ j +(A x B y A y B x ) ˆ k | A × B| = AB sin θ = A B = AB (direction via right-hand rule) Quadratic: ax 2 + bx + c =0 x 1,2 = b ± b 2 4ac 2a Derivatives: d dt (at n )= nat n1 d dt sin at = a cos at d dt cos at = a sin at Integrals: t2 t1 f (t)dt = a n+1 (t n+1 2 t n+1 1 ) if f (t)= at n , then f (t)dt = a n+1 t n+1 + C (n = 1) sin at dt = 1 a cos at cos at dt = 1 a sin at Kinematics: translational rotational v= r2 r1 t2t1 v = d r dt a= v2v1 t2t1 a = dv dt = d 2 r dt 2 r(t)= r 0 + t 0 v(t ) dt v(t)= v 0 + t 0 a(t ) dt ω= θ2θ1 t2t1 ω = dt α= ω2ω1 t2t1 α = dt = d 2 θ dt 2 θ(t)= θ 0 + t 0 ω(t ) dt ω(t)= ω 0 + t 0 α(t ) dt —– constant (linear/angular) acceleration only —– r(t)= r 0 + v 0 t + 1 2 at 2 v(t)= v 0 + at v 2 x = v 2 x,0 +2a x (x x 0 ) (and similarly for y and z) r(t)= r 0 + 1 2 (v i + v f )t θ(t)= θ 0 + ω 0 t + 1 2 αt 2 ω(t)= ω 0 + αt ω 2 f = ω 2 0 +2α(θ θ 0 ) θ(t)= θ 0 + 1 2 (ω i + ω f )t Energy and Momenta: translational rotational K = 1 2 Mv 2 W = F · d r const −−−→ force F · Δ r P = dW dt = F · v p cm = m 1 v 1 + m 2 v 2 + ... = Mv cm J = F dt p F ext = Ma cm = d p cm dt F int =0 if F ext,x =0,p cm,x = const τ = r × F and | τ | = F r = Fl K rot = 1 2 I tot ω 2 W = τdθ const −−−→ torque τ Δθ P = dW dt = τ · ω L = r × p = I 1 ω 1 + I 2 ω 2 + ... = I tot ω τ ext = I tot α = d L dt τ int =0 if τ ext,z =0,L z = const —– Work-energy and potential energy —– W K E tot,i + W other = E tot,f U = F · d r ; U grav = Mgy cm ; U elas = 1 2 kΔx 2 F x (x)= dU (x)/dx F = U = ∂U ∂x ˆ i + ∂U ∂y ˆ j + ∂U ∂z ˆ k Constants/Conversions: g = 9.80 m/s 2 = 32.15 ft/s 2 (Earth, sea level) 10 m/s 2 33 ft/s 2 1 mi = 1609 m 1 lb = 4.448 N 1 ft = 12 in 0.454 kg (Earth, sea level) 1 in = 2.54 cm 1 rev = 360 =2π radians Circular motion: a rad = v 2 R a tan = d| v| dt = T = 2πR v s = v tan = Relative velocity: v A/C = v A/B + v B/C v A/B = v B/A Forces: Newton’s: F = ma, F B on A = F A on B Hooke’s: F x = kΔx friction: | f s |≤ µ s |n|, | f k | = µ k |n| Centre-of-mass: r cm = m 1 r 1 + m 2 r 2 + ... + m n r n m 1 + m 2 + ... + m n (and similarly for v and a) Phys 206 — Exam III Formulae

others
• Category

## Documents

• view

47

0

### Transcript of Phys206—ExamIIIFormulaemechanics.physics.tamu.edu/formulae/phys206-examIII-formulae.pdfTrigonometryandVectors:...

Trigonometry and Vectors:

sin 30◦ = cos 60◦ = 1

2sin 36.9◦ ≈ cos 53.1◦ ≈ 3

5

sin 45◦ = cos 45◦ = 1√2

sin 53.1◦ ≈ cos 36.9◦ ≈ 4

5

sin 60◦ = cos 30◦ =√3

2

hadj = h cos θ = h sinφ

hopp = h sin θ = h cosφ

opp

tan θ =hopp

hhopp

φ

θ

Law of cosines: C2 = A2 +B2 − 2AB cos γ

Law of sines:sinα

A=

sinβ

B=

sin γ

C

αC

A

β

~A = Axi+Ay j +Az k A =~A

| ~A|~A · ~B = AxBx +AyBy +AzBz = AB cos θ = A‖B = AB‖

~A× ~B = (AyBz−AzBy )i+ (AzBx−AxBz)j + (AxBy−AyBx)k

| ~A× ~B| = AB sin θ = A⊥B = AB⊥ (direction via right-hand rule)

ax2 + bx+ c = 0 ⇒ x1,2 =−b±

√b2 − 4ac

dt (atn) = natn−1

ddt sin at = a cos atddt cos at = −a sin at

Integrals: ∫ t2t1

f(t)dt = an+1

(tn+1

2 − tn+1

1 )if f(t) = atn, then

{

f(t)dt = an+1

tn+1 + C(n 6= −1)

sin at dt = −1

a cos at∫

cos at dt = 1

a sin at

Kinematics:translational rotational

〈~v〉 = ~r2−~r1t2−t1

~v = d~rdt

〈~a〉 = ~v2−~v1

t2−t1~a= d~v

dt =d2~rdt2

~r(t) = ~r0 +∫ t

0~v(t′) dt′

~v(t) = ~v0 +∫ t

0~a(t′) dt′

〈ω〉 = θ2−θ1t2−t1

ω = dθdt

〈α〉 = ω2−ω1

t2−t1α= dω

dt =d2θdt2

θ(t) = θ0 +∫ t

0ω(t′) dt′

ω(t) = ω0 +∫ t

0α(t′) dt′

—– constant (linear/angular) acceleration only —–

~r(t) = ~r0 + ~v0t+1

2~at2

~v(t) = ~v0 + ~at

v2x = v2x,0 + 2ax(x− x0)(and similarly for y and z)

~r(t) = ~r0 +1

2(~vi + ~vf )t

θ(t) = θ0 + ω0t+1

2αt2

ω(t) = ω0 + αt

ω2f = ω2

0 + 2α(θ − θ0)

θ(t) = θ0 +1

2(ωi + ωf )t

Energy and Momenta:

translational rotational

K = 1

2Mv2

W =∫

~F · d~r const−−−→force

~F ·∆~r

P = dWdt = ~F · ~v

~pcm = m1~v1 +m2~v2 + . . .

= M~vcm

~J =∫

~Fdt = ∆~p

∑ ~Fext = M~acm =d~pcmdt

∑ ~Fint = 0

if∑

Fext,x = 0, pcm,x = const

~τ = ~r × ~F and |~τ | = F⊥r = Fl

Krot =1

2Itotω

2

W =∫

τdθconst−−−→torque

τ ∆θ

P = dWdt = ~τ · ~ω

~L =∑

~r × ~p

= I1~ω1 + I2~ω2 + . . .

= Itot~ω

~τext = Itot~α =d~L

dt∑

~τint = 0

if∑

τext,z = 0, Lz = const

—– Work-energy and potential energy —–

W = ∆K Etot,i +Wother = Etot,f

U = −∫

~F · d~r ; Ugrav = Mgycm ; Uelas =1

2k∆x2

Fx(x) = −dU(x)/dx ~F = −~∇U = −[

∂U∂x i+

∂U∂y j +

∂U∂z k

]

Constants/Conversions:

g = 9.80 m/s2= 32.15 ft/s

2(Earth, sea level)

≈ 10 m/s2 ≈ 33 ft/s

2

1 mi = 1609 m

1 lb = 4.448 N 1 ft = 12 in⇔ 0.454 kg (Earth, sea level) 1 in = 2.54 cm

1 rev = 360◦ = 2π radians

Ratan =

d|~v|dt

= Rα

T =2πR

vs = Rθ vtan = Rω

Relative velocity: ~vA/C = ~vA/B + ~vB/C

~vA/B = −~vB/A

Forces:Newton’s:

∑ ~F = m~a, ~FB on A = −~FA on B

Hooke’s: Fx = −k∆x

friction: |~fs| ≤ µs|~n|, |~fk| = µk|~n|Centre-of-mass:

~rcm =m1~r1 +m2~r2 + . . .+mn~rn

m1 +m2 + . . .+mn

(and similarly for ~v and ~a)

Phys 206 — Exam III Formulae

Moments of inertia:

Thin cylinder

I = MR2

Thin cylinder

I = 1

2MR2 + 1

12ML2

Hollow sphere

I = 2

3MR2

Solid sphere

I = 2

5MR2

Solid cylinder

I = 1

2MR2

Solid cylinder

I = 1

4MR2 + 1

12ML2

Thin rod

I = 1

12ML2

Thin rod

I = 1

3ML2

Thick cylinder

I = 1

2M(R2

1 +R22)

Thick cylinder

I = 1

4M(R2

1+R22) +

1

12ML2

Rectangular plate

I = 1

12M(a2 + b2)

Rectangular plate

I = 1

3Ma2

For a point-like particle of mass M a distance R from the axis of rotation: I = MR2

Parallel axis theorem: Ip = Icm +Md2

A disk is a cylinder of negligible length; the I for a disk may be found by setting L = 0 in the formulae for cylinders