ME   212   –   Dynamics   –   Final   Study   Guide
ME   212   –   Dynamics
Final   Study   Guide
Compiled   by   Frank   Owen
Mechanical   Engineering,   Cal   Poly,   San   Luis   Obispo
Good   engineering   practice
Leave   numerical   quantities   out   of   an   analysis   as   long   as   possible,   until   the   last   step,   so   that   you   can   see   how   an   analysis’s   quantities   interact   and   what   rôles   they   play.
Do   a   unit   analysis   of   every   result,   even   those   without   numerical   quantities,   to   make   sure   that   the   result   is   unit ‐ consistent.
Pick   either   the   SI   or   the   English   system—it   doesn’t   matter.
If   the   units   are   consistent   in   one   system,   they   will   be   in   the   other   one   too.
Every   time   you   write   an   equilibrium   equation   or   a   momentum ‐ balance   equation,   preface   that   equation   with   a   statement   of   what   the   equation   is   (for   example,   → ∑   ),   including   the   sign   convention   for   the   equation.
Friction
With   Coulomb   friction,   the   static   friction   relationship   is   an   inequality:
∙
It   is   a   very   common   mistake   amongst   Dynamics   students   to   use   the   relationship   ∙ ,   even   if   there   is   not   impending   motion/slip.
It   must   be   stated   that   this   is   the   case   before   you   can   use
Vectors   this   relationship   between   F f
and   N .
If   there   is   sliding   motion   between   two   bodies   at   a   contact   surface,   F f
=
 k
N .
This   is   an   equality,   not   an   inequality,   so   the   relationship   doesn’t   have   the   problems   that   the   relationship   F f
≤
 s
N   has.
Never   ever   write   a   vector   equation   without   arrows   over   the   vector   quantites.
Never   ever   write   a   vector   equation   that   mixes   vector   and   scalar   terms.
That   is,   you   cannot   add   or   subtract   a   scalar   and   a   vector.
Vector   division   is   meaningless;   it   does   not   exist.
The   multiplicands   of   a   cross   product   must   both   be   vectors.
Know   how   to   perform   vector   cross   products   with   unit   vectors.
The   derivative   of   a   vector   with   respect   to   time   can   be   either   a   change   in   magnitude   of   the   vector   over   time   or   a   change   in   direction   of   the   vector   over   time.
Or   it   can   be   both   together.
So   there   can   be
(normal)   acceleration,   even   if   the   speed   of   a   particle   or   point   stays   the   same.
Know   how   to   construct   a   unit   vector   in   a   specific   direction.
(For   example,   what   is   the   unit   vector   for   the   direction

degrees   past   180   degrees,   measured   CCW   from   the   positive   x   axis?)    Another   useful   way   to   construct   a   unit   vector   in   the   direction   of   a   known   vector   is   simply   to   divide   that   known   vector   by   its   magnitude.
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ME   212   –
Particle   kinematics
Rectilinear   motion
and      ;   from   these   come
Dynamics   –   Final   Study
∙    and
Guide
∙   .
Know   how   to
If   a   set   up   these   definite   integrals,   their   limits   of   integration,   and   then   how   to   do   the   integration.
For   the   above   integrals,   need      and       to   perform   the   integration.
is   constant,    ∙   ,    and   ∙ ∙   .
There   is   also   the   funny   relationship   ∙ ∙   .
This   can   be   used   if   a   is   constant,    if   a   =   a ( x ),    if    v   =   v ( x ),    if   a   =   a ( v ).
In   all   of   these   cases,   collect   functions   of   like   variables   on   one   side   of   the   equation   and   integrate   them   using   the   proper   integration   limits.
Planar   motion
Cartesian   coordinates   ( x / y )   –   Projectile   motion:    constant   acceleration   in   the   y   direction   ( a y
=   ‐ g )    and   a x
=   0.
Polar   coordinates   ( r /

)   –   I   like   the   derivation   of   the   expression   for     in   polar   coordinates.
One   starts   with   the   displacement   vector    ∙ ̂    and   uses   the   product   rule   and   some   drawings   to   get      in   polar   coordinates.
The   rules   of   differentiation   yield   Coriolis   acceleration,   for   example.
Path   coordinates   ( n / t )   –   Velocity   always   in   t   direction.
Acceleration   in   t   direction   represents   increasing   or   decreasing   speed.
Acceleration   in   the   n   direction   represents   a   change   in   direction   of   the   velocity   vector.
So   in   a   car,   the   accelerator   and   the   brake   are   the      controls.
But   the   steering   wheel   is   an   accelerator   too.
It’s   the      control.
Particle   kinetics
N2L
FBD=MAD   –   Start   with   this   equation,   then   draw   the   diagrams,   leaving   out   any   part   of   the   environment   touching   the   object   being   analyzed.
W/E
A   hint   to   use   W/E   on   a   kinetics   problem   is   if   a   situation   is   described   with   distances   covered   between   two   end   states.
→
Don’t   calculate   the   work   of   gravity   or   a   spring.
Account   for   these   changes   in   energy   state   as    ∆    and
∆
.
Be   able   to   make   power   calculations   too.
∙
,
∙
.
I/M
A   hint   to   use   I/M   on   a   kinetics   problem   is   if   a   situation   is   described   with   elapsed   time   given   between   two   end   states   or   if   beginning   and   end   v s   or
 s   are   given   or   sought.
Linear   and   angular   momentum   conserved   if

F   or

M   =0.
Angular   momentum   of   a   particle   about   a   point   is   just   moment   of   linear   momentum   about   that   point.
Without   spatial   extension,
0
.
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ME   212   –   Dynamics   –   Final   Study   Guide
e   =   0   is   plastic   impact,   so   bodies   stick   together   ( )
e   =   1   is   elastic   impact.
Energy   is   conserved   during   impact.
Impact
Impact   defines   n   and   t   directions.
For   non ‐ rough   impact   (all   that   we   covered),   no   impulse   forces   in   tangential   direction.
So   no   change   in   pre ‐  and   post   impact   tangential    velocities.
Change   in   normal   velocities   depends   on   coefficient   of   restitution   ( e ).
Rigid ‐ body   kinematics
Rotary   analogues   to   1 ‐ D   motion
and      ;   from   these   come   ∙    and    ∙   .
If     is   constant,   the   constant ‐ acceleration   relationships   are   analogous   to   the   case   of   1 ‐ D   motion.
There   is   also   the   funny   relationship    ∙ ∙    and   all   the   interesting   variations   of   integration   that   go   with   it   (see   “Rectilinear   motion”   above).
Relative   motion
Know   why   you   write   the   subscript   of

AB
and

AB
without   a   “/”   and   why   v
A/B
and   a
A/B
have   a   “/”.
Know   what   the   “/”   means.
If   a   body   is   “released   from   rest”,   it   is   not   rotating,   so   there   are   no   normal   accelerations   (see   Pledge   of   Allegiance   below).
On   a   rigid   body,   there   is   only   tangential   velocity   between   any   two   points—A   and   B.
B   cannot   approach   A,   nor   can   it   recede   from   A.
The   distance   between   A   and   B   never   changes.
If   the   reference   point   is   A,   then   on   a   rotating   body   AB,   it   looks   to   an   observer   on   A   that   point   B   is   just   moving   left   or   right.
It   looks   as   if   B   is   rotating   around   A.
Even   though   B   cannot   approach   A,   it   can   accelerate   toward   A.
This   is   just   the   normal   acceleration   of   circular   motion   toward   the   apparent   center   of   rotation.
So
/
always   points   from   B   to   A.
For   relative   velocity   and   relative   acceleration,   I   really   recommend   using   velocity   and   acceleration   diagrams   to   graphically   illustrate   the   vector   sums
/
and
/ /
.
Know   a   mechanism,   like   a   4 ‐ bar   or   a   slider ‐ crank   real   well.
Knowing   one   well   carries   over   to   other   mechanisms.
The   Pledge   of   Allegiance
/
/
/
/
/ / /
/ /
/ /
This   is   true   for   any   two   points   on   a   rigid   body   in   general   plane   motion.
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ME   212   –   Dynamics   –   Final   Study   Guide
Rigid ‐ body   kinetics
is   radius   of   gyration   about   center   of   mass.
It   gives     :
.
N2L   other   than   the   center   of   mass,   and   such   a
∙ .
k   can   be   given   about   points
See   particle   N2L.
On   the   MAD,   draw   all     vectors   and     at   the   center   of   mass.
To   sum   moments,   if   the   problem   is   to   find   accelerations   at   the   instant   shown   (often   a   release   from   rest),   you   can   sum   moments   about   any   point.
Choose   a   point   strategically,   to   eliminate   as   many   unknown   forces   as   possible   from   the   moment   equilibrium   equation.
W/E
If   multiple   bodies   constrained   to   move   together,   just   take   the   total   energy   of   each   “before”,   the
I/M   total   work   done   on   them   or   by   them   (usually   friction)   during   the   motion,   and   then   the   total   energy   of   each   “after”.
Be   able   to   draw   a   momentum   diagram   and   write   the   momentum   balance   for   a   problem.
Draw   all   s   and   s   of   and   at   the   center   of   mass.
Impulse   part   of   diagram   will   have   all   forces   and   moments   integrated   by   t .
Preface   all   equations   related   to   momentum   diagram   with
∑
,
∑
,
∑
to   show   what   you   are   doing   and   include   sign   convention.
Impact
Use   momentum   diagram   for   these   analyses.
The   relationship   for   e   is   the   same,   but   the   velocities   in   the   formula   apply   to   the   velocities   of   each   body   at   the   point   of   impact   (POI).
If   a   body   strikes   an   immovable   barrier,   probably   don’t   know   direction   of   impact   force   from   obstacle   onto   body,   so   probably   will   sum   momentum   moments   about   the   POI   to   eliminate   the   contact   force   from   the   I/M   equation.
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