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3 ´u= du F=G Mm {G =6.67 ×10−11 m } 2 r dt s kg r A → B (B w . r . t A)=r B =r B−r A A

dv a ( s)=v =¿ ads=vdv ds Rectilinear Motion:

B

]

dr =´v B =´v B−´v A → v´ B=´v A + v´ B dt A A dv a= = a´ B =´a B−´a A → a´ B= a´ A + a´ B dt A A v=

∑ F´ x =m a x : ∑ F´ y =m a y : ∑ F´ z =m az

Normal & Tangential 2

∑ F´ t=m a t=m ´v : ∑ F´ n=m an=m vρ : ∑ F´ b=0 s=rθ →ds=rdθ → vdv=ads =at rdθ v

θ

v0

θ0

∫ vdv=∫ a t rdθ Cylindrical

∑ F´ r=m a r=m ( ´r −r θ´ 2) ´ r´ θ´ ) [ corriollis acc 2 r´ θ´ ] ∑ F´ θ=m aθ= m ( r θ+2 ∑ F´ z =m az =m ( z´ )

1 s= a t t2 + v 0 t +s 0 2 v =´s =a t t+ v o 2 2 v =v 0 +2 at (s− s0 ) a´ =v´ u´ t +u´´ t v=at u´ t +an u´ n 2

v an = (centripital Acc) ρ

( )

dy 2 32 ⌋ dx

| | 2

d y d x2

Cylindrical

r x =r cos θ , r y =r sin θ [θ rad ] , z=z ,

rad ] v r =´r , v θ=r θ´ , vz=´z ´θ=angular vel[ s ar =r´ −r θ´ 2 , a θ=r ´θ+2 r´ θ´ , az = z´ Relative Motion: Independent Translating

( )

−1

∅=tan

Normal & Tangential

ρ=

A

Kinetics: Cartesian

1 2 s x =x=s x 0 + v x 0 t + a t 2 1 s y = y =s y 0 +v y0 t+ a t2 2 v x 0=v 0 cos θ , v y 0 =v 0 sin θ v x = s´x =at+ v x0 , v y = s´y =at+ v y 0 ´ x´ = v´ , a y = s´y = y´ = v´y ax= sx= x u |u|= √ ux +u y ;θ=tan−1 y ux

⌊ 1+

A

2 S B +h+ S A =l T S´ A + 2´S B=0=¿ 2 v B =−v A 2 aB =−a A

Curvilinear Motion: Cartesian

s´ = ρθ´

A

Dependent

1 2 s= a t +v 0 t+s 0 2 ds v = =at +v 0 dt dv a= dt 2 2 v =v 0 +2 a(s −s 0 )

at =´v =´s ,

[

´r B =´r A + r´ B → r´ B =r´ B− r´ A , ´r B=− r´ A

r dr dθ

3 )=θ 100 dU =F´ ∙ d ´r = F´ ds cos ( θ ) [ N ∙ m ] { +→∈direction of action tan −1 (

Work: Note 3% slope=

r2

s2

[Variable F ] U 1 →2=∫ F´ ∙ d r´ =∫ F cos (θ ) ds r1

s1 s2

[Constant Force straight ] U 1 → 2=F c cos ( θ ) ∫ds s1 z2

[Weight :+ ↓ ] U 1 →2=∫−wdz=−w ( z 2− z 1 ) z1 s2

[Spring ]U 1→ 2=∫ (−ks ) ds= s1

−1 2 2 k (s 2 −s 1 ) 2

Principle of Work and Energy (Forces on particle expressed in terms of displacement)

m at ds=mvdv → s2

v2

1

1

[

1 ∫ ∑ F´ t ds=∫ mvdv= 2 m (v 22−v21) →U 1 → 2=T 2−T 1 T = v s T 1 +U 1 → 2=T 2[ initial k . e+work=Final k . e ]

Power:

P=

∆U J d r´ ´ = F ∙ ´v =Fv cos θ watt= = N ∙ m = F´ ∙ s dt ∆t s

[

Efficiency:

]

Impact: Stages 1.)

Before Impact ( t < t 0) =initial momentum for particale 2.)

power ouput energy output = ε= energy input power input

t2

Deformation Impulse I AB (effect of A on B )=∫ P A dt , t1

Potential Energy: Gravitational

t2

m A v a 1−∫ Pdt =m A v

V g =wh [ above datum ]=− wh[ below datum ]

t1 t2

Elastic:

1 2 V e= k s 2 V´ =V´ g + V´ e ´ g 1+ V´ e1 ) − ( V ´ g 2+ V´ e2 ) U 1 →2= V´ 1−V´ 2= ( V

t1

3.)

t2

t1

4.) Restitution Impulse

Conservation of Energy:

t2

m A (v)−∫ Rdt =m A v A 2

T U 1−2 ∑¿ ¿ (¿ ¿ 1+V 1)+¿ ¿ T 1 +( V e1 +V g 1 ) =¿

t1 t2

I R=∫ Rdt=m A (v ) −m A ( v A 2) : I D >I R t1

´L A 2=m v´ A 2 : L´ B 2 =m v´ B 2 5.) After Impact Conservation of linear momentum

T 2 +( V e2 +V g 2 )

Disp Vel

Rectilin angular S ϴ V ω

Acc

A

α

Cartesian

cylindrical

(x , y , z ) (vx , v y , vz ) (a x , a y , az )

(r ,θ , z ) ( vr , v θ , v z ) (a r , aθ , az )

Principle of Impulse and Momentum: (Forces on particle expressed in terms of time)

d ´v d (m v´ ) ´ ´ = a=m ∑ F=m dt dt t2

t1

L A + LB =L AB m A ( v A 1) + m B ( v B 1) =m A( v A 2 )+ m B (v B 2 ) t2

∫ Pdt=m A ( v A 1 )−m A (v) t1 t2

∫Rdt=m A ( v ) −mA ( v A 2 ) t1

Coefficient of restitution

∫ Rdt = v−v A 2 ∫ Pdt v A 1 −v ∫ Rdt = v B 2−v e B= ∫ Pdt v −v B 1 e A=

v2

t2

∑ F´ dt =d ( m v´ ) →∫ ∑ F´ dt=∫ md v´ =∫ ∑ F´ dt=m v´ v1

t1

t2

m v´ 1+∫ ∑ F´ dt=m v´ 2 → ´L1+ ´I 1 → 2=L´ 2

e=

t1 t2

v B 2−v A 2 Relative velocity of seperation = v A 1 −v B 1 Relative velocity of approach

Angular Momentum and Impulse

´I =∫ ∑ F´ dt [Linear impulse ] t1

]

´L=m v´ linear momentum acts∈´v (kg ∙ m ) s Cartesian:

[ L1´i+ L1 ´j+ L1 k´ ]+ [I ´i+ I ´j + I k´ ]=[ L2 ´i+ L2 ´j + L2 ´k ] t2

m v x 1 +∫ F x dt=m v x2 (same format for y , z) t1

Maximum Deformation

I M =∫ Pdt= ( m A +m B ) v

´ g 2+ V´ e2 ) − ( V ´ g 1+ V´ e1 ) U 2 →1= V´ 2−V´ 1= ( V

[

∫ Pdt =m A ( v A 1 )−m A (v)

I D =¿

particle∈ x . y planeabout point O (z axis) ¿ = H [Scalar Form] ( O) z (r )( mv ) [ r is moment arm ] ´ O= ´r × m ´v [Vector Form] H ´ O=¿ H ´´ moment about O=time rate change H M O ∑¿

´ ´ FO =¿ LO resultant force = time rate change of L ∑¿ t2

t2

´ ) dt ´ O dt=∫ (r´ × F angular impulse∫ M t1

t1

a´ A → acceleration of base point A ω ´ → angular velocity of body α´ → angular acceleration of body r´ B → position vector directed ¿ A ¿ B A

Principle of Angular Impulse and Momentum

´ O )2 ( H´ O) 1+∑∫ M O dt= ( H ∑ ( H´ O )1=∑ ( H´ O ) 2 [conservation of ang momentum] Planar kinematics of Rigid Body Translation [position]

r´ B =´r A +´r B A

[velocity] v´ B=´v A [acceleration] a´ B=´a A Rotation (about a fixed axis)

[angular position ] θ= 1 α t2 +ω 0 t +θ0

2 dθ [angular velocity ] ω= =αt +ω 0 dt 2 2 ω =ω0+2 α (θ−θ0 ) dω [angular acceleration ] α= : αdθ =ωdω dt (for the point P from point O)

´v B=v´ A +

(a P )t =αr , ( aP )n=ω r ( ´ ) = ´ ´ ( ´ ) +=´Ω´ × ´( ´ ´ rB / A) v´ B=´v A +( v´ B / A ) xyz (

)

2 →a´ P = ( α´ × r´ P / O )−ω r´ P / O

General plane (relative motion analysis)ri

r´ B =´r A +´r B A

(

´ ´ [Velocity] v´ B=´v A +´v B =´v A + ω× r B

A

)

v´ B → velocity of B v´ A → velocity of base point A ω ´ → angular velocity of body r´ B → position vector directed ¿ A ¿ B

v´ B → absolute velocity of B measured ¿ XYZ v´ A → velocity of origin A of x , y , z referance measured ¿ the XYZ referance ( v´ B / A) xyz → velocity of B w . r . t A observed ¿ the x , y , z referance (Ω´ ) →insitanious angular velocity of moving referance frame x , y , z r´ B / A → position vector of B w . r .t A [acceleration]

A

[Acceleration]

´ (v´ B/ A ) xyz+ (a´ B/ A ) xyz ´ × ´r B / A) ] +2 Ω× a´ B=´a A +(Ω´´ ×´r B/ A ) +[ ´Ω×( Ω

a´ B=´a A +( a´ B / A ) t +( a´ B / A ) n a´ B=´a A +( α´ ×´r B/ A) + [ ω× ´ ( ω×´ ´ r B/ A )] ´a B=a´ A +( α´ × r´ B/ A ) −ω 2 r´ B A

a´ B → acceleration of B

d r´ B / A dt

d r´ B / A = ´v B / A +( Ω´ × r´ B / A ) dt

2

A

r´ B =´r A + r´ B / A ^ y ^j r´ B / A=x Bi+ B [velocity]

s P =θr , v P=v O +v P /O =v O +ωr , v´ P= v´ O + v´ P /O =´v O + ω ´ × r´ P/ O

[Position]

Relative Motion Analysis using Rotating Axes [Position]

a´ B = acceleration of B, measured from the X, Y, Z reference

a´ A

= acceleration of the origin A of the x, y, z reference, measured from the X, Y, Z reference (a´ B / A )xyz ,( v´ B / A) xyz=¿ acceleration and velocity of B with respect to A, as measured by an observer attached to the rotating x, y, z reference

´´ , ´ = angular acceleration and angular velocity of Ω Ω the x, y, z reference, measured from the X, Y, Z reference r´ B / A = position of B with respect to A...