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Equation sheet to be used on exams...

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Formula Sheet, ECE105

1

Mathematics

Rotational motion:

Work:

2

quadratic equation, ax + bx + c = 0 : √ −b ± b2 − 4ac x= 2a cosine law: c 2 = a2 + b2 − 2ab cos θ

vectors:

θ ω vr α at

= = = = =

ar

=

s/r dθ/dt ωr dω/dt αr v2 r

kinematics of uniform angular acceleration:

~ ~·B A ~ ~×B A

= =

AB cos α AB sin α

ωf

=

ωi + α∆t

θf

=

θi + ωi ∆t +

ω f2

=

ωi2 + 2α∆θ

Linear momentum:

Fundamental Constants

= =

Σ~ τext

=

uniform linear acceleration a = const: =

vis + as ∆t

sf

=

si + vis ∆t +

v 2fs

=

1 as (∆t)2 2

vis2 + 2as ∆s

vAD = vAB + vBC + vCD

moment of inertia: I = mr

2

or

Ihoop = mr 2

Newton’s Laws: 1 ~ 1 dP Fnet = m m dt ~ A on B = −F~B on A F

Second:

~a =

Third:

friction:

Ibody =

P

2 i mi ri

2 mr 2 5 Parallel axis theorem: Isphere =

1 Idisk = mr 2 2 Ithin

rod

=

1 ml2 12

I = Icm + M d2 Angular momentum:

Fs = −k∆s fs fk

≤ =

µs FN µk FN

~ L

= mgy

Us

2 = 21 k (∆s)

K

=

Kr

=

= ~r × ~ p = Iω

1 mv 2 2 1 Iω 2 2

(gravitational potential) (spring potential) (translational kinetic) (rotational kinetic)

Waves and Sound ω = 2πf

k=

1 kx2 2

2π λ

x = A cos φ = A cos (ωt + φ◦ ) v = −Aω sin (ωt + φ◦ ) a = −Aω 2 cos (ωt + φ◦ ) = −ω 2 x

Travelling wave: y = A psin(kx ∓ ωt + φ◦ ) with v = λ/T = λf = T/µ Superposition of two  waves traveling in the  same direction: 2A cos (∆φ/2) sin kxavg − ωt + (φ◦ )avg

~r × F~ ~ τnet /I ~ dL/dt

about an axis through the geometrical centre:

Relative Velocity:

Ug

f = 1/T

Torque:

vfs

Energy:

(

Impulse:

~τ ~ α

(if F~ is constant) (if F~ is conservative)

simple harmonic oscillator, F = −kx, U =

~ ∆t = ∆~ p J~ = F

Mechanics

= Fs ds ~ · ∆~r =F = −∆U

~ p = m~v

g = 9.80 m s−2

Hooke’s Law:

1 α (∆t)2 2

dW

(particle) (rigid body)

Superposition of two waves traveling in opposite directions: 2A sin (kx) cos (ωt) Standing waves: A(x) = 2a sin kx λm

=

2L , where m = 1, 2, 3... (fixed ends) m...