Equation Sheet for ES – DynamicsMiscellaneous

If ax2 + bx+ c = 0, then x =−b±

√b2 − 4ac

2a

d

dxsinx = cos x, d

dxcos x = − sinx∫

1

ax+ bdx =

1

aln(ax+ b)

Rectilinear ( -D) Motion

Position: s(t), Velocity: v = s = ds/dt, Acceleration: a = v = s = d2s/dt2 = vdv/ds. Whenthere is constant acceleration ac

v2 = v20 + 2ac(s− s0) v = v0 + act s = s0 + v0t+1

2act

2

For varying acceleration

v(t) = v0 +

∫ t

t0

a(t)dt, s(t) = s0 + v0(t− t0) +

∫ t

t0

[∫ t

t0

a(t)dt

]dt

ere are equivalent equations for constant angular acceleration α.Projectile Motion

x(t) = v0t cos β y(t) = v0t sin β − 1

2gt2

D Motions — Cartesian Coordinates

Position: r = xı+yȷ Velocity: v = dr/dt = xı+ yȷ Acceleration: a = dv/dt = d2r/dt2 = xı+ yȷ

D Motions — Normal-Tangential (Path) Coordinates

v = vut a = vut +v2

ρun

D Motions — Polar Coordinates

r = rur v = rur + rθuθ a = (r − rθ2)ur + (rθ + 2rθ)uθ

D Motions — Cylindrical Coordinates

r = rur + zuz v = rur + rθuθ + zuz a = (r − rθ2)ur + (rθ + 2rθ)uθ + zuz

D Motions — Spherical Coordinates

r = rur v = rur + rϕuϕ + rθ sinϕuθ

a = (r−rϕ2+rθ2 sin2 ϕ)ur+(rϕ+2rϕ−rθ2 sinϕ cosϕ)uϕ+(rθ sinϕ+2rθ sinϕ+2rϕθ cosϕ)uθ

Relative Motion

rB = rA + rB/A vB = vA + vB/A aB = aA + aB/A

Newton’s Second Law

F = ma, Fx = max, Fy = may, Fn = man, Ft = mat, Fr = mar, Fθ = maθ

Work-Energy Principle

U1−2 =

∫path

F · dr T =1

2mv2 T1 + U1−2 = T2

Vg = mgh Ve =1

2kδ2 T1 + V1 + (U1−2)nc = T2 + V2

Linear Impulse-Momentum Principle

p = mv F = ˙p mv1 +

∫ t1

t2

F dt = mv2

Angular Impulse-Momentum Principle

h0 = rP/O ×mvP MO =˙hO hO1 +

∫ t2

t1

MOdt = hO2

Impact of Smooth Particles

n indicates LOI direction, t indicates tangent direction

m1(v−1 )n+m2(v

−2 )n = m1(v

+1 )n+m2(v

+2 )n, (v−1 )t = (v+1 )t, (v−2 )t = (v+2 )t, e =

(v+2 )n − (v+1 )n(v−1 )n − (v−2 )n

System of Particles∑F = maG T1 + V1 + (U1−2)nc = T2 + V2

T =1

2mv2G +

N∑i=1

1

2miv

2i/G h = mvG∑

F = ˙p∑

MG =˙hG∑

MP =˙hG + rG/P ×maG

∑MP = (

˙HP )rel + rG/P ×maP

Mass Flows

F = mf (vB − vA) mf = ρAQA = ρBQB

Q = vS MP = mf (rD/P × vB − rC/P × vA)

F = ma+ movo − mivi

Rigid Body Kinematics

vB = vA + vB/A aB = aA + aB/A

vB = vA + ω × rB/A aB = aA + α× rB/A + ω × (ω × rB/A)

aB = aA + α× rB/A − ω2rB/A

Rotating Reference Frames

vP = vA + vPrel+ Ω× rP/A

aP = aA + aPrel+ 2Ω× vPrel

+˙Ω× rP/A + Ω×

(Ω× rP/A

)Rolling Without Slip

ω = −v0R

α = −a0R

Moments of Inertia

Disk IG =1

2mr2 Rod IG =

1

12ml2

Plate IG =1

12m(a2 + b2) Sphere IG =

2

5mr2

Ring IG = mr2 Cylinder IG =1

2mr2

Parallel Axis eorem: IA = IG +md2, Radius of Gyration: IA = mk2A

Angular Momentum and Equations of Motion for a Rigid Body

Angular MomentumHG = IGω HO = IGω + rG/O ×mvG

Equations of MotionTranslation motion equations are given by ∑

F = maG

Rotational motion equation. For the mass center G:∑

MG = IGα and for the xed point O:∑MO = IOα In the case of an arbitrary point A,∑

MP = IGα+ rG/P ×maG∑

MP = IP α+ rG/P ×maP (MA)FBD = (MA)KD

Work-Energy for a Rigid Body

e work-energy principle is the same as the principle for particles. e kinetic energy of a rigidbody is

T =1

2IOω

2 T =1

2mv2G +

1

2IGω

2

Rigid Body Impact

linear portion where n indicates along LOI and t is tangent to LOI

m1(v−1 )n+m2(v

−2 )n = m1(v

+1 )n+m2(v

+2 )n, (v−1 )t = (v+1 )t, (v−2 )t = (v+2 )t e =

(v+2 )n − (v+1 )n(v−1 )n − (v−2 )n

Angular portion, y is the distance from the center of mass to the impact point

e =(v+2 )n + ω+

2 y2 − (v+1 )n − ω+1 y1

(v−1 )n + ω+1 y1 − (v−2 )n − ω+

2 y2, (h−

O)1 = (h+O)1 (h−

O)2 = (h+O)2

Constrained Impact(h−

O)1 + (h−O)2 = (h+

O)1 + (h+O)2