MEC311%Exam%Formula%sheet%W2015%
#
1.
Kinematics%
#Particle%translating%/%rotating%about%a%fixed%axis%in%constant%acceleration%motion:%
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%
s = avoo + act
ω = ωo + αct
s = so + svot + ½ act2
2
2t
sv2 ==(svoo)2++ 2ac (s – so)
θ = θo + ωot + ½ αct2
ω2 = (ωo)2 + 2αc (θ – θo)
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Translating%body:%%% v A = vB ,%% a A = a B %
%
General%plane%motion:%%
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•
•
•
•
•
%
•
•
%
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#
#
#
#
!
!
!
!
relationship%of%angular%motion:% ! (t) = d! (t) / dt ,% ! (t) = d! (t) / dt %
s! = r!!, !!
s = r!!! %%
projectile%motion:%ahoriz%=%0,%%avert%=%Lg%
!
v2
acceleration%in%nLt%components:% a = v" û t + û n %%
!
relationship%of%two%points%A%and%B%using%fixed%frame:%

 

 
v B = v A + v B / A = v A + ω × rB / A %
!
! !
!
!
!
!
!
!
a B = a A + a B/A = a A + ! AB " rB/A + # AB " (# AB " rB/A ) %
!
!
!
!
! !
For%planar%motion:% a B = a A + ! AB " rB/A # $ 2AB rB/A %
!
!
" û %%
polar%coordinates:%velocity:% v = r"û + r!" û ,%acceleration:% a = (""r ! r"" 2 ) û + (r""
" + 2 r"")
r
r
!
"
position,%velocity%and%acceleration%in%Cartesian%coordinates:%
2. Newton’s%Second%Law##
(G:%mass%center.%A:%point%of%fixed%axis%of%rotation.%O:%any%point%moment%is%calculated%about.)%
Translation%
FixedLaxis%
General%plane%motion%of%
rotation%of%
symmetric%bodies%
symmetric%bodies%
%
%
!
!
!
!
!
!
F = maG %
!
F
=
m
a
F
=
m
a
%
%
!
!
G
G
!
!


!
!
M
!
G = I G! ,%or%%
M
=
I
!
%
%
M
=
0
!
A
A
∑ G


∑ M o = Ioα %
%
%
%
%
Mass%moment%of%inertia%%%I = ∫ r2 dm = ∫ r2ρdV.%%%Io = Ix+ Iy.%%%ParallelLaxis%theorem%I = IG + md2
Friction:%Impending%slip:%F=µsN;%slip:%F=µkN%
Force%in%the%spring:%Fs=kδ=k(LLL0)%
Page%1%of%3%
3.%Work–Energy%method%
%
Principle%of%work%and%energy% %
% T1 + V1 + U1−2 = T2 + V2 %
%
Kinetic%Energy:%
%
FixedLaxis%rotation%of%
symmetric%bodies%
T=
General%plane%motion%of%
symmetric%bodies%
1
I Aω 2 %
2
T=
1
1
mVG2 + I Gω 2 %
2
2
%

  
%%%%%%%%%%%Work%Done:% U1→2 = ∫ ( F • dr + M • dθ ) %
2
1
Energy%conservation:% T1 + V1 = T2 + V2 %
1 2
kδ %
2
1
%%%%%%%%%%Work%of%a%spring%force:% (U1−2 )e = − k(δ22 − δ12 ) %%
2
%%%%%%%%%Power:%P=dU/dt;%efficiency:%ε=(power%output)/(power%input)%
%
%
%%%%%%%%%%Potential%Energy:% V g = mgh %%% Ve =
4.%Impulse–Momentum%method%
"
!
Linear%momentum pi = mv i , i = 1,2,… #
 

Angular%momentum H i = r × mvi ,i = 1,2,…
#
Impulse–Momentum%Principle%
%
#
FixedLaxis%rotation%of%symmetric%
bodies%
General%plane%motion%of%
symmetric%bodies%
2 


m( vG )1 + ∑ ∫ FK dt = m( vG ) 2 %
2 


m( vG )1 + ∑ ∫ FK dt = m( vG ) 2 %



( I Aω )1 + ∑ ∫ M Adt = ( I Aω ) 2 %
2 


( I Gω )1 + ∑ ∫ M G dt = ( I Gω ) 2
t
K t1
t2
l
t1
t
K t1
t
l
%
t1
%
t2
Linear%impulse:%

% ∑ FK dt %%%%%
∫
K t1
t

! !
Conservation%of%linear%momentum: If% ∑ FK dt = 0, %%% p1 = p 2 %
2
∫
K t1
Angular%impulse:% ∑ M dt %
t2
l
∫
G
t1
Angular%Momentum%about%an%axis%passing%through%any%point%A:% H A = I G ! + (d)(mv G ) %%
%
Page%2%of%3%
If%point%A%is%instantaneous%center%of%zero%velocity:% H IC = IICϖ %
Conservation%of%angular%momentum
t
!
!

If% ∑ ∫ M G dt = 0, H1 = H 2 %
2
l
t1
Central%Impact:%%%%!!! !!! + !! !!! = !! !!! + !! !!!
Coefficient%of%restitution%in%the%line%of%impact:%%%! =
Oblique%impact%coefficient%of%restitution:%
!=
Radius of curvature
!!! !!!!
!!! !!!!
!!"! − !!"!
!!"! − !!"!
%
5.% Mass%moment%of%inertia:%
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Page%3%of%3%