Equation Sheet ka4o

Equation Sheet for E MCH 212 — Dynamics Miscellaneous If ax2 + bx + c = 0, then x = −b ±

√

. b2 − 4ac 2a. Law of Cosines: C 2 = A2 + B 2 − 2AB cos θ.

For two planar orthogonal coordinate systems with unit vector pairs (ˆ ua , u ˆb ) and (ˆ uc , u ˆd ), the first system can be written in of the second using: u ˆa = (ˆ ua · u ˆc ) u ˆc + (ˆ ua · u ˆd ) u ˆd

and

u ˆb = (ˆ ub · u ˆc ) u ˆc + (ˆ ub · u ˆd ) u ˆd

Elementary (1-D) Motions Position: q(t); Velocity: v = q˙ = dq/dt; Acceleration: a = q¨ = dv/dt = d2 q/dt2 = vdv/dq. Z t Z t Z t Given a = a(t): v(t) = v0 + a(t) dt q(t) = q0 + v0 (t − t0 ) + a(t) dt dt t t0 t0 Z 0v Z v 1 v Given a = a(v): t(v) = t0 + dv q(v) = q0 + dv a(v) a(v) v0 v Z q Z q0 dq Given a = a(q): v 2 (q) = v02 + 2 a(q) dq t(q) = t0 + v(q) q0 q0 Rectilinear (1-D) Motion Position: s(t); Velocity: v = s˙ = ds/dt; Acceleration: a = s¨ = dv/dt = d2 s/dt2 = vdv/ds. For constant acceleration ac (the second line are the corresponding rotation equations): v 2 = v02 + 2ac (s − s0 )

v = v0 + ac t

s = s0 + v0 t + 12 ac t2

ω 2 = ω02 + 2αc (θ − θ0 )

ω = ω0 + αc t

θ = θ0 + ω0 t + 12 αc t2

General Motions—Time Derivative of a Vector ~ = Au ~ A|, ~ and letting ω ~ Given A ˆA , with u ˆA = A/| ~ A be the angular velocity of the vector A: u ˆ˙ A = ω ~ ×u ˆA

~˙ = A˙ u ~ A ˆA + ω ~ ×A

~¨ = A¨ u ~+ω ~ A ˆA + 2~ ωA × A˙ u ˆA + ω ~˙ A × A ~A × ω ~A × A

2D Motions—Cartesian Coordinates/Components Position: ~r = x ˆı + y ˆ;

Velocity: ~v = d~r/dt = x˙ ˆı + y˙ ˆ;

Acceleration: ~a = d~v /dt = d2~r/dt2 = x ¨ ˆı + y¨ ˆ

2D Motions—Normal-Tangential Components ~v = v u ˆt

˙ v = ρ|θ|

~a = at u ˆt + an u ˆn

at = v˙

an = ρθ˙2 = v 2 /ρ

2D Motions—Polar Coordinates/Components ~rr = r u ˆr ~a = ar u ˆr + aθ u ˆθ

~v = vr u ˆ r + vθ u ˆθ 2 ˙ ar = r¨ − rθ

vr = r˙

vθ = rθ˙

aθ = rθ¨ + 2r˙ θ˙

Newton-Euler Equations F~ = m~a;

Fx = max ;

Fy = may ;

Fn = man ;

Ft = mat ;

Fr = mar ;

Fθ = maθ

Last modified: December 11, 2010

Balance of Work & Energy Z U1-2 = F~ · d~r

T = 21 mv 2

path

Vg = W y = mgy

T1 + U1-2 = T2 Ve = 21 kδ 2

T1 + V1 + (U1-2 )nc = T2 + V2

Balance of Impulse & Momentum t2

Z

d(m ~v ) F~ = = p~˙ dt

F~ dt = m~v2 − m~v1

t1

Impact of Smooth Particles For a coefficient of restitution e, in the direction along to the line of impact (LOI) or n direction: + v + − vA vseparate e= = B − − vapproach vA − v B

Total linear momentum is conserved in the n direc+ + − − tion: (mA vA +mB vB = mA vA +mB vB )n . Velocity of each particle is conserved perpendicular to the LOI + + − − (t direction): (vA = vA )t and (vB = vB )t .

Angular Impulse-Momentum Principle ~hP = ~rQ/P × m~vQ

~ P = ~h˙ P + ~vP × m~vQ M

Z

t2

~r × F~ dt =

t1

Z

t2

~ P dt = ~hP 2 − ~hP 1 M

t1

Rigid Body Kinematics ~vB = ~vA + ~vB/A = ~vA + ω ~ body × ~rB/A 2 ~aB = ~aA + ~aB/A = ~aA + α ~ body × ~rB/A + ω ~ body × (~ ωbody × ~rB/A ) = ~aB = ~aA + α ~ body × ~rB/A − ωbody ~rB/A

Moments of Inertia (IG )disk = 12 mr2

(IG )rod =

2 1 12 ml

(IG )plate =

2 1 12 m(a

+ b2 )

(IG )sphere = 52 mr2

2 Parallel Axis Theorem: IA = IG + md2 ; Radius of Gyration: IA = mkA

Equations of Motion for a Rigid Body The general equations of motion for a rigid body are given by F~ = m~aC and by the following equations for the moments about an arbitrary point P , which is on the rigid body: ~ P = IG α M ~ + ~rG/P × m~aG

or

~ P = IP α M ~ + ~rG/P × m~aP

~ G = IG α If you sum moments about the mass center G, then M ~. Work-Energy for a Rigid Body The work-energy principle is the same as that for particles. The kinetic energy of a rigid body is: 2 T = 12 mvG + 21 IG ω 2 (G is the mass center) Unit Conversions and Miscellaneous 1 mi = 5280 ft;

g = 32.2 ft/s2 = 9.81 m/s2 ;

m (slugs) = [W (pounds)]/g

GmA mB F~on A = u ˆB/A 2 rAB

Last modified: December 11, 2010