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Force,&Mass&&&Acceleration&Investigation&
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Introduction:!Using!the!spring!scale!attached!to!a!cart,!you!will!create!a!constant!force!as!you!move!the!cart!back!
and!forth!across!your!lab!table.!!!
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1. Predict!how!you!will!need!to!move!the!cart!in!order!to!create!this!constant!force.!!Discuss!with!your!table!
and!write!what!you!expect!will!happen.!
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Have!each!member!of!the!group!try!to!keep!the!cart!moving!with!a!constant!force.!
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2. Does!the!motion!match!your!prediction?!!Describe!how!you!keep!the!force!constant.!
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3. Describe!the!motion!of!the!cart!when!you!keep!a!low!constant!force!on!it.!
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4. Describe!the!motion!of!the!cart!when!you!keep!a!high!constant!force!on!it.!
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5. Keep!a!constant!force!of!0.5!N!on!the!spring!scale!and!describe!how!the!speed!changes!as!your!cart!moves!1!
meter.!Try!a!few!times!until!you!get!results!that!seem!consistent.!!Begin!and!end!the!motion!as!smoothly!as!
possible.!!Discuss!with!your!group!and!write!your!observations!below.!
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6. Have!the!same!member!of!the!group!do!the!same!experiment!with!a!constant!force!of!2!N.!!Is!this!more!or!
less!difficult?!!Why?!!!
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7. Discuss!with!your!group!and!come!up!with!a!qualitative!observation!of!the!relationship!between!change!in!
velocity!(acceleration)!and!amount!of!force.!
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8. Have!a!different!member!of!the!group!keep!the!force!constant!at!0.5!N!with!zero!masses,!2!masses!and!then!
4!masses.!!How!did!the!difference!in!mass!affect!the!difficulty!of!keeping!the!force!constant?!
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9. Discuss!with!your!group!and!come!up!with!a!qualitative!observation!of!the!relationship!between!change!in!
velocity!(acceleration)!and!amount!of!mass.!
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10. Qualitatively,!how!would!you!assume!acceleration!relates!to!both!mass!and!force!in!the!same!
proportionality?!!(a!proportionality!is!similar!to!an!equation,!but!since!you!don’t!know!the!relationship!for!
sure,!you!use!a!proportionality!symbol!instead!of!an!equal!sign.!!It!looks!like!alpha,!or!like!an!open!infinity!∝)!
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What&was&Newton's&big&idea?&
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1. Think!about!rolling!a!ball!down!an!incline.!Describe!the!motion!and!how!the!velocity!changes!in!detail!
beginning!when!you!are!holding!it!still!at!the!top!of!the!incline!to!when!it!finally!hits!the!bottom.!
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2. Imagine!there!is!no!friction!and!the!ball!goes!back!up!an!identical!incline.!!When!will!it!stop?!!Draw!an!image!
of!the!initial!condition!(just!before!you!let!go!of!the!ball)!and!the!final!condition!(when!it!stops)!below.!
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3. What!if!the!incline!is!no!longer!identical!and!the!opposite!side!is!not!as!steep?!How!high!will!the!ball!roll!on!
the!less!steep!incline?!Draw!the!initial!and!final!conditions!below.!!
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4. Take!away!the!second!incline!altogether.!How!long!will!the!ball!continue!to!move!on!this!frictionless!plane?!!
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5. Aristotle!believed!there!was!a!link!between!force!and!velocity;!he!thought!that!there!was!a!link!between!the!
amount!of!force!applied!on!an!object!and!how!fast!it!would!move.!How!is!the!answer!to!#4!different!from!
Aristotle's!claim?!
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6. When!you!push!on!an!object,!what!relationship!with!motion!do!you!expect?!If!you!push!harder,!what!
happens?!
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Dynamics&
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Kinematics!taught!us!to!distinguish!between!accelerated!and!un]accelerated!motion.!!!
But!what!causes!acceleration?!!!
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Example)!Luca!pushes!on!a!chair!with!wheels,!assume!it!is!essentially!frictionless.!
1. Describe!how!the!motion!of!the!chair!changes!as!he!continues!to!push!on!the!chair.!!
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2. What!if!he!pushes!harder?!!How!will!the!motion!of!the!chair!change?!!
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3. What!if!he!keeps!his!push!constant!so!he!is!always!applying!the!same!amount!of!"push"!to!the!chair?!
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4. What!happens!to!the!motion!of!the!chair!when!Saaya!goes!to!the!other!side!of!the!chair!and!also!pushes!on!
it?!!How!can!you!mathematically!describe!how!the!chair!will!move!when!there!are!two!people!pushing!on!it?!!
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5. What!happens!when!she!pushes!with!the!same!amount!of!"push"!as!Luca?!
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In!order!to!change!the!motion!of!an!object,!a!"net"!force!must!be!applied,!and!this!"net"!force!causes!the!
acceleration,!which!then!changes!the!object’s!velocity.!!The!net!force!is!proportional!to!the!resulting!acceleration,!
and!the!proportionality!constant!for!a!given!object!is!its!mass.!!The!greater!the!mass,!the!more!force!is!needed!to!
impart!an!acceleration!on!the!object.!!Newton's!second!law!(the!first!is!a!consequence!of!the!second)!states!this!in!
the!familiar!equation:!
∑ F = ma ,!where!by! ∑ F !we!mean!"net"!force,!which!is!the!vector!sum!of!all!the!forces!on!
the!mass!m.!
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In!the!example!from!the!previous!page,!what!would!be!the!mass?!!
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What!would!be!the!sum!of!forces?!!
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Write!out!Newton's!second!law!for!our!example!and!solve!for!the!acceleration!of!the!chair.!
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Newton's!third!law,!often!called!the!law!of!"action!and!reaction,"!is!very!important!and!often!misunderstood.!!
It!states!that!forces!always'occur'in'pairs,!and!that!each!force!of!a!pair!acts!on!a!different'object.!!If!object!A!exerts!a!
force!on!object!B,!then!B!exerts!an!equal!but!opposite!force!on!A.!!If!I!push!hard!on!the!wall!of!a!building,!it!doesn't!
appear!to!exert!any!force,!but!it!does.!!It!pushes!equally!hard!on!me.!!But!remember!that!"it"!and!"me"!are!two!
different!objects,!so!the!forces!don't!cancel.!!
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Example:!In!an!accident,!an!18!wheeler!truck!collides!with!a!Smart!car.!Which!of!the!two!feels!more!force?!Explain.!
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Remember!Newton's!3rd!Law:!for!every!force!there!is!an!equal!and!opposite!force.!Knowing!this,!what!is!the!
difference!between!the!force!the!car!feels!and!the!force!the!truck!feels?!!
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Imagining!this!scenario,!we!know!that!the!car!would!be!much!more!affected!by!this!collision!than!the!truck.!Using!
Newton's!2nd!Law,!compare!the!forces,!masses!and!accelerations!felt!by!each!vehicle.!What!aspect!of!this!equation!
explains!why!the!car!is!totaled!and!the!truck!is!fine?!
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We!will!start!our!investigation!of!dynamics!by!identifying!a!few!of!the!forces!we!will!be!working!with.!
1.
Drop!a!pen!from!your!hand!to!the!floor.!!Think!carefully!about!the!time!that!the!pen!is!falling.!
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a) What!is!the!acceleration!of!the!pen!as!it!falls?!
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b) As!you!learned!in!the!introductory!activity,!if!something!is!accelerating,!then!there!must!be!a!“net”!force!on!
it.!!Use!Newton’s!2nd!law!to!write!down!the!equation!that!describes!the!force,!mass!and!acceleration!of!the!
pen!as!it!falls.!!
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c) What!is!the!common!name!of!the!force!acting!on!the!pen!that!
causes!it!to!accelerate!downward?!!!
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d) What!causes!that!force?!
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e) Is!this!force!acting!when!the!pen!is!still!in!your!hand?!
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f) Draw!a!simple!picture!of!the!pen!as!a!single,!small!box!or!a!dot!in!the!space!below.!!!
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g) Draw!a!vector!on!the!box!or!dot!to!represent!the!force!or!
forces!on!the!pen!as!it!falls!from!your!hand.!
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h) Label!the!force.!
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A!picture!like!this!is!called!a!free!body!diagram.!!It!includes!only!a!system!of!interest,!and!the!forces!acting!
on!it.!!Our!system!above!is!just&the&pen.!!In!Newtonian!physics,!all!forces!are!considered!as!arising!from!an!
interaction!between!two!objects.!!Forces!are!specified!by!identifying!the!object!on'which!the!force!is!exerted!and!the!
object!that!is!exerting!the!force.!!For!example,!the!gravitational!force!above!is!exerted!on&the&pen&by&the&Earth.!!The!
other!forces!we!use!in!our!FBDs!(free!body!diagrams)!are!the!tension!force,!the!normal!force!and!the!applied!force.!
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The!Tension!force!is!applied!by!a!
Forces
string!or!a!rope,!an!example!is!an!
object!hanging!from!a!rear!view!
Non-Contact
Contact
mirror!in!a!car.!!We!assume!all!
strings/ropes!are!massless!in!this!
Gravity
Surfaces
Strings
class.!This!force!is!labeled!T.!
(others)
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The!Normal!force!is!a!surface!contact!
Sliding
Pulling
Pushing
force,!this!force!is!why!your!hand!
Normal (⊥ )
Friction (||)
Tension
does!not!fall!through!the!table!when!
you!touch!it.!This!force!is!always!
perpendicular!to!the!point!of!contact!–!in!fact!the!word!"normal"!in!geometry!means!a!vector!that!is!perpendicular!
to!a!plane.!!This!force!is!labeled!N.!!
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Any!force!not!belonging!to!gravity,!normal!or!tension!is!simply!called!an!Applied!Force!and!is!labeled!F.!
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i) Looking!at!the!box,!can!you!tell!the!direction!of!the!acceleration!on!the!pen?!Explain!using!Newton's!2nd!Law.!
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If!the!pen!has!a!mass!of!0.05!kg,!what!is!the!force!acting!on!it?!(you!can!use!10!m/s2!for!g)!
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Practice&Problems!
1.!For!each!of!the!examples!that!follow,!draw!a!free]body!diagram!in!the!space!provided.!
a)!!A!book!is!at!rest!on!a!table.!!Draw!a!picture!of!the!scenario!and!diagram!the!forces!on!the!book.!!
Write!out!Newton's!second!law!for!the!book.!
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b)!!A!baseball!is!freefalling!through!the!air.!!Diagram!the!forces!on!the!ball.!!Ignore!air!resistance.!!
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Again,!write!out!Newton's!second!law.!!
What!is!the!direction!of!the!acceleration?!
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c)!!A!slanted!surface!has!two!blocks!on!it!as!shown.!!They!are!attached!by!a!rope.!!
Diagram!the!forces!on!both!masses,!individually.!!Draw!them!below,!not!on!the!image.!
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d)!!Now!two!masses!are!attached,!but!one!is!on!the!slanted!surface!
and!one!is!hanging.!!Draw!the!free!body!diagrams!for!each!mass!
individually!below.!
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Redraw!the!FBD!for!the!slanted!mass!but!this!time!change!the!forces!so!that!they!all!align!on!one!of!2!perpendicular!
axes.!Choose!one!of!the!axes!as!the!direction!of!acceleration!of!the!block.!
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e)!!You!are!pulling!a!sled,!which!is!on!the!ground,!with!a!rope!that!is!held!at!a!35°!angle!to!the!ground.!!Draw!a!
picture!of!the!scenario!on!the!left,!and!then!diagram!the!forces!on!the!sled!on!the!right.!
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f)!Redraw!a!free!body!diagram!of!this!same!sled!scenario,!but!this!time!change!the!forces!so!that!they!all!align!on!
one!of!2!perpendicular!axes.!Choose!one!of!the!axes!as!the!direction!of!acceleration!of!the!sled.!
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2.!!Carefully!consider!the!forces!acting!on!a!1!kg!mass!hanging!from!a!
string.!!
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a) Draw!force!vectors!on!the!box!to!the!right!to!represent!these!two!
forces.!Label!these!forces!on!the!diagram.!
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b) Write!down!the!magnitude!of!this!force!or!forces,!include!units.!!
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c) What!is!the!name!of!the!force!that!is!preventing!this!mass!from!falling?!Which!direction!does!this!force!have!
to!be!acting!to!prevent!the!1!kg!mass!from!falling?!Explain!your!choice!using!Newton's!2nd!Law.!!
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a) Is!this!second!force!strong!enough!to!make!the!mass!rise!up!into!the!air?!!
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b) Compare!the!direction!of!the!force!preventing!the!mass!from!falling!to!the!direction!of!the!string.!!!Are!they!
in!the!same!direction,!perpendicular,!parallel,!etc?!
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c) How!do!the!magnitudes!of!these!two!forces!compare?!!Are!they!the!same,!is!one!much!larger,!etc.?!!
How!do!you!know!this!–!explain!using!Newton's!2nd!Law.!
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3.!!Consider!a!situation!where!you!can!choose!the!system!for!your!FBD!based!on!what!information!you!need.!
Here!we!have!two!masses!being!pushed!by!a!student.!Draw!the!different!FBDs!indicated!for!the!masses.
C
A
B
A) Adam pushes on box A, which pushes box B
Free-body diagram for box A
Free-body diagram for box B
Free-body diagram for system C
consisting of both boxes
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Applications&of&Newton's&Laws&
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Applying!Newton's!Laws!involves!the!following!steps:!!(an!alternate!method!here)!
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Isolate!the!system.!
The!system!is!what'is'accelerating'or'could'accelerate.!Sometimes!you!may!have!to!choose!more!than!one!system!to!
solve!a!problem.!
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2.!
Draw!a!free8body'diagram.!
This!is!an!abstract,!simplified!representation!of!the!system,!with!arrows!representing!each!of!the!forces!exerted!on!
the!system!by!the!environment.!!!
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3.!
Choose!axes!and!resolve!forces!along!them.!
It!is!always!wise!to!choose!the!directions!of!your!axes!so!that!the!net!force!
trigonometry!to!resolve!forces!along!your!chosen!axes,!if!necessary.!
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4.!
Often!applying!
Apply!
∑ F !points!along!one!of!them.!!Then!use!
∑ F = ma !along!each!of!your!chosen!axes.!
∑ F = ma !along!one!axis!will!give!you!the!equation!you!need!to!solve!the!problem,!but!you!may!
also!need!to!do!so!along!two!axes!and!solve!two!equations!with!two!unknowns.!
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As!you!become!familiar!with!these!steps!you!may!find!shortcuts,!however,!as!you!are!starting!out!I!recommend!you!
go!through!this!process!methodically!to!avoid!confusion.!
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Unit&3:Force& &
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Solved&Example&
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A!4!kg!block!is!pulled!along!a!frictionless!surface!by!a!10!N!force!at!an!angle!of!37°!to!the!horizontal.!
A) What!is!the!acceleration!of!the!block?!
0N
6N
1
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37°
1. The!system!is!the!block.!
4
kg
2. See!free]body!diagram!to!the!right.!
3. Choose!x]axis!to!be!horizontal,!y]axis!vertical.!!(Since!the!angle!is!37°,!the!components!
of!the!10!N!vector!form!a!3]4]5!triangle.)!
y
N
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∑ Fx = max
10
4. !!!
B)! What!is!the!normal!force!of!the!surface!on!the!block?!
NBS
( +8! N ) = ( 4 kg )( a )
37°
F
=
ma
∑ y
y
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8N
x
( N + 6 N − W ) = (0)
WBE
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W = mg = ( 4 kg ) 10 m/s2 = 40 N
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N = 40 N − 6 N = 34 N
Practice&Problems!
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1.!
A!20!kg!block!hangs!from!a!string!tied!to!the!ceiling,!and!a!10!kg!block!hangs!from!another!string!tied!to!the!
first!block.!
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T1
a) Draw!FBDs!for!each!mass.!Label!all!forces.!
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20 kg
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b) What!is!the!tension!in!the!upper!string?!Sum!all!forces!using!Newton's!2nd!Law.!! 10 kg 10 kg
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c) What!is!the!tension!in!the!lower!string?!!
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d) What!would!the!tensions!be!if!the!blocks!were!reversed?!
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A!4!kg!block!is!pulled!horizontally!along!a!frictionless!surface!by!a!10!N!force!at!an!angle!of!0°!to!the!
horizontal.!!
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a) Draw!a!free!body!diagram!below!the!image.!
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b) What!is!the!acceleration!of!the!block?!
4 kg
4 kg
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FBD!
c) What!is!the!normal!force!of!the!surface!on!the!block?!
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3.!
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A!4!kg!block!is!pulled!along!a!frictionless!surface!by!a!10!N!force!at!an!angle!of!37°!above!the!horizontal.!!
a) Draw!a!free!body!diagram!of!the!block.!
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b) What!is!the!acceleration!of!the!block?!
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10 N
37°
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kg
c) What!is!the!normal!force!of!the!surface!on!the!block?!
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4.!
A!5!kg!block!rests!on!a!frictionless!inclined!plane,!tied!by!a!string!to!the!wall.!!The!incline!is!at!an!angle!of!37°!
to!the!horizontal.!
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a)
Draw!a!FBD!of!the!block.!!
5
kg
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37°
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b)
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What!is!the!tension!in!the!string?!
c)
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What!is!the!normal!force!of!the!plane!on!the!block?!!
d)
If!the!string!were!to!be!cut,!what!would!the!acceleration!of!the!block!be?!
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e)
Answer!part!C)!in!the!general!case!of!a!mass!m!on!an!incline!of!angle!θ!]!solve!the!problem!again!using!only!
variables!.!!
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5.!
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Substitute!θ!=!0°!and!θ!=!90°!into!your!answer!to!part!d)!–!does!your!answer!make!sense?!Explain.!!
A!passenger!with!a!mass!of!80!kg!stands!on!the!floor!of!an!elevator!of!mass!250!kg!as!it!accelerates!upward!
with!an!acceleration!of!3!m/s2.!!
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a)!!
Draw!3!FBDs:!the!passenger,!the!elevator!and!both!of!them!together.!!!
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b)!!
What!is!the!tension!in!the!elevator!cable?!!
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What!is!the!normal!force!of!the!elevator!floor!against!the!
passenger?!
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d)!
Answer!parts!b)!and!c)!if!the!elevator!is!accelerating!down!at!3!m/s2.!!
Answer!parts!A)!and!B)!if!the!elevator!is!moving!up!at!a!constant!speed!of!3!m/s.!
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Frictional!forces!come!in!two!kinds.!!If!the!surfaces!are!actually!sliding!against!each!other,!the!force!is!called!
a!kinetic!frictional!force.!!If!the!surfaces!are!not!actually!sliding,!but!only!attempting!to!slide,!the!force!is!called!a!
static!frictional!force.!!Usually!the!static!frictional!force!is!stronger!than!the!kinetic!force!for!the!same!surfaces.!
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Coefficient&of&Kinetic&Friction&
1. What!is!easier:!pushing!a!mass!across!a!field!of!grass!or!across!ice?!Why?!
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2. When!you!roll!a!pen!without!the!cap!across!the!desk,!what!stops!it!from!rolling?!
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3. When!you!push!your!packet!across!the!desk,!what!stops!it!from!continuing!to!move?!
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For!two!given!surfaces,!the!kinetic!frictional!force!fk!is!found!to!be!proportional!to!the!normal!force!N!that!pushes!
the!surfaces!together.!!The!proportionality!constant!is!a!dimensionless!number!called!the!coefficient'of'kinetic'
friction,!symbolized!by!the!Greek!letter!µ!(myew).!!So:!
fk = µ k N !
Coefficient&of&Static&Friction!
Place!your!calculator!on!the!desk!in!front!of!you.!Lightly!push!it,!but!don't!move!it.!Continue!gradually!increasing!
your!force!until!it!moves.!!
1. Draw!a!free!body!diagram!of!when!you!pushed!on!it!with!very!little!force!on!the!left,!then!on!the!
right!draw!a!free!body!diagram!of!when!you!pushed!on!it!with!enough!force!to!almost!make!it!move.!
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2. What!was!the!acceleration!in!each!case?!
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3. Friction!kept!the!calculator!from!moving,!describe!how!the!frictional!force!changed!as!you!kept!
pushing!harder!and!harder.!!
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4. Can!friction!be!greater!than!the!force!of!your!hand?!What!would!that!look!like?!
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Static!friction!is!a!"smart"!force;!it!changes!depending!on!circumstances,!up!to!a!limit.!!That!maximum!static!
frictional!force!is,!like!the!kinetic,!proportional!to!the!normal!force!N,!and!the!proportionality!constant!is!called!the!
coefficient'of'static'friction.!!We!write!this!relation!as:!
fsMAX = µ s N !
MAX
The!actual!static!frictional!force!fs!varies!from!zero!to! fs
MAX
slip,!then!the!static!frictional!force!is! fs
and!no!more.!
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Practice&Problems& &
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1.!
!depending!on!the!situation.!!If!the!surfaces!are!about!to!
.!!If!not,!then!the!static!frictional!force!is!just!enough!to!prevent!slipping,!
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A!4!kg!block!is!on!a!rough!horizontal!surface!with!coefficients!of!friction!as!shown!below.!!It!is!
pushed!by!a!10!N!force!as!shown.!!!
Applied!Force!
µk = 0.1
4 kg
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µs = 0.2
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a)!Draw!an!FBD!of!the!mass.!
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b)!Does!the!block!move?!!!
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c)!If!so,!what!is!its!acceleration?!
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13!
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2.
The!same!block!is!pushed!with!a!10!N!force!at!a!30°!angle!above!
the!horizontal,!on!the!same!surface.!!!
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10 N
a)!Draw!an!FBD!of!the!mass.!
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30°!
37°
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µk = 0.1
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µs = 0.2
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b)!Does!the!block!move?!!If!so,!what!is!its!acceleration?!
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A!5!kg!block!is!on!a!rough!(frictional)!37°!incline!with!coefficients!of!friction!as!shown!below.!!!
a)!Draw!an!FBD!of!the!mass.!
5!k
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µk!=!0.1!
µS!=!0.2!
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b)!!After!the!string!is!cut,!will!the!block!slide!down?!!!
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c)!!If!so,!what!is!its!acceleration?!!!
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37°
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d)!!Would!the!situation!change!if!the!mass!is!increased?!Explain.!
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14!
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A!block!of!mass!m1!is!connected!by!a!string!over!
a!pulley!to!a!mass!m2.!!The!surface!of!the!table!
has!coefficients!of!!
friction!µs!and!µk.!
a) Draw!FBDs!of!both!masses.!
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b)!
What!is!the!minimum!value!of!µs!to!prevent!slipping?!!
c)!
When!the!blocks!slip,!find!the!acceleration!of!the!blocks!using!variables!only.!!
d)!
When!the!blocks!slip,!find!the!tension!in!the!string!using!variables!only.!
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For!this!same!situation,!assume!mass!1!is!8!kg!and!mass!2!is!2!kg.!The!coefficient!of!static!friction!is!0.2!
and!the!coefficient!of!kinetic!friction!is!0.1.!!!
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e)!!
Find!the!acceleration!of!the!system,!if!it!is!accelerating.!!
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f)!!
Find!the!tension!in!the!string.!!
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5.!! Blocks!A,!B,!and!C!are!connected!by!strings!2!and!3,!and!are!pulled!across!a!frictionless!table!by!string!1!as!
shown.!!Blocks!B!and!C!are!identical,!and!each!has!twice!the!mass!of!block!A.!!Assume!that!the!strings!are!
massless!and!horizontal.!
A)! Draw!free]body!diagrams!in!the!spaces!provided.!
Free-body diagram for block A
Free-body diagram for block B
Free-body diagram for block C
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B)! Rank!the!magnitudes!of!the!accelerations!of!blocks!A,!B,!and!C!(aA,!aB,!and!aC).!!Explain.!
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C)! Rank!the!magnitudes!of!the!net'forces!on!blocks!A,!B,!and!C.!!Explain.!
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D)! Rank!the!magnitudes!of!the!tensions!in!strings!1,!2,!and!2!(T1,!T2,!and!T3).!!Explain.!
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E)! If!mass!A!is!5!kg!and!the!force!being!applied!is!50!N,!what!is!the!acceleration!of!the!system?!
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F)! What!is!the!tension!in!string!3?!
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16!
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