Dynamics and Newton's Laws
Introduction to Dynamics
- Dynamics is the study of what causes motion.
- Kinematics, studied previously, describes how objects move using plots and equations.
- Mechanics, the overarching field, is split into kinematics and dynamics.
- Dynamics will cover forces (now) and energetics (later).
Newton's Three Laws of Motion
Newton's First Law (Law of Inertia)
- Statement: In an inertial reference frame, a body remains in a state of uniform motion (constant velocity and direction) unless acted upon by a net external force.
- Uniform motion implies constant velocity (both magnitude and direction) or being at rest (v=0).
- Inertial Reference Frame: A reference frame that is not accelerating.
- Examples: A stationary observer on the ground (approximately), a car moving at a constant velocity.
- If working in a non-inertial frame, the approach must be modified (e.g., using fictitious forces).
- Non-Inertial Reference Frame: An accelerating reference frame.
- The surface of the Earth technically is a non-inertial frame due to rotation, but for many problems, it can be approximated as inertial.
- Example: Ice Block on an Accelerating Truck
- Observer (You) on the Ground (Inertial Frame):
- Observes the truck accelerate.
- Observes the ice block slide off the back of the truck (assuming no friction).
- Explanation: From your perspective, the ice block has an initial velocity and no net external force acts on it. According to Newton's First Law, it maintains its initial velocity while the truck accelerates forward beneath it, causing it to appear to slide backward.
- Observer (Friend) on the Truck (Non-Inertial Frame):
- Observes the ice block accelerate away from them (sliding off the back).
- From the friend's perspective, they are stationary relative to the truck, and yet the ice block accelerates, suggesting a net force acting on the ice block.
- Resolution: To make Newton's laws work in this accelerating frame, one must introduce a pseudo force or fictitious force. This fictitious force points in the direction that explains the observed acceleration of the ice block from within the truck's frame.
Newton's Second Law
- Formal Statement: The net (or sum of) external force acting on an object is equal to the product of the object's mass and its acceleration.
- Mathematically: egin{aligned} ext{Sum of forces external} &= m imes a \ ext{Or } ext{sum of } oldsymbol{ ext{F}}{ ext{ext}} &= moldsymbol{a} ext{ (Vector form)}\ ext{Or } oldsymbol{ ext{F}}{ ext{net}} &= moldsymbol{a} ext{ (Common notation)}\ ext{Where } m &= ext{mass of the object} ext{ (units: kg)}\ oldsymbol{a} &= ext{acceleration vector} ext{ (units: m/s}^2 ext{)}\ oldsymbol{ ext{F}}{ ext{net}} &= ext{net force vector} ext{ (units: N)}\ ext{(N = Newton)}\ ext{One Newton (N)} &= 1 ext{ kg} imes 1 ext{ m/s}^2 ext{ (Derived SI unit)}\ ext{To measure a Newton}: ext{A 1 kg object given a 1 m/s}^2 ext{ acceleration by a push has a pushing force of 1 N.} ag{1}\ ag{2} ag{3} ag{4} ag{5}\ ag{6} ag{7}\ ag{8} ag{9} ag{10} ag{11} ag{12} ag{13} ag{14} ag{15} ag{16} ag{17} ag{18} ag{19} ag{20} ag{21} ag{22} ag{23} ag{24} ag{25} ag{26} ag{27} ag{28} ag{29} ag{30} ag{31} ag{32} ag{33} ag{34} ag{35} ag{36} ag{37} ag{38} ag{39} ag{40} ag{41}\ ag{42} ag{43} ag{44} ag{45} ag{46} ag{47} ag{48} ag{49} ag{50}\ ag{51} ag{52} ag{53} ag{54} ag{55} ag{56} ag{57} ag{58} ag{59} ag{60} ag{61} ag{62} ag{63} ag{64} ag{65} ag{66} ag{67} ag{68} ag{69} ag{70} ag{71} ag{72} ag{73} ag{74} ag{75} ag{76} ag{77} ag{78} ag{79} ag{80}\ ag{81} ag{82} ag{83} ag{84} ag{85} ag{86} ag{87} ag{88} ag{89} ag{90} \ ag{91} ag{92} ag{93} ag{94} ag{95} ag{96} ag{97} ag{98} ag{99} ag{100} ag{101} ag{102} ag{103} ag{104} ag{105} ag{106} ag{107} ag{108} \ ag{109} ag{110} ag{111} ag{112} ag{113} ag{114} ag{115} ag{116} ag{117} ag{118} ag{119} ag{120} ag{121} ag{122} ag{123} ag{124} ag{125} ag{126} ag{127} ag{128} ag{129} ag{130} ag{131} ag{132} ag{133} ag{134} ag{135} ag{136} ag{137} ag{138} ag{139} ag{140} ag{141} ag{142} ag{143} ag{144} ag{145} ag{146}} ag{147} ag{148} ag{149} ag{150} ag{151} ag{152} ag{153} ag{154} ag{155} ag{156} ag{157} ag{158} ag{159} ag{160} ag{161} ag{162} ag{163} ag{164} ag{165} ag{166} ag{167} ag{168} ag{169} ag{170} ag{171} ag{172} ag{173} ag{174} ag{175} ag{176} ag{177} ag{178} ag{179} ag{180} ag{181} ag{182} ag{183} ag{184} ag{185} ag{186} ag{187} ag{188} ag{189} ag{190} ag{191} ag{192} ag{193} ag{194} ag{195} ag{196} ag{197} ag{198} ag{199} ag{200} ag{201} ag{202} ag{203} ag{204} ag{205} ag{206} ag{207} ag{208} ag{209} ag{210} ag{211} ag{212} ag{213} ag{214} ag{215} ag{216} ag{217} ag{218} ag{219} ag{220} ag{221} ag{222} ag{223} ag{224} ag{225} ag{226} ag{227} ag{228} ag{229} ag{230} ag{231} ag{232} ag{233} ag{234} ag{235} ag{236} ag{237} ag{238} ag{239} ag{240} ag{241} ag{242} ag{243} ag{244} ag{245} ag{246} ag{247} ag{248} ag{249} ag{250} ag{251} ag{252} ag{253} ag{254} ag{255} ag{256} ag{257} ag{258} ag{259} ag{260} ag{261} ag{262} ag{263} ag{264} ag{265} ag{266} ag{267} ag{268} ag{269} ag{270} ag{271} ag{272} ag{273} ag{274} ag{275} ag{276} ag{277} ag{278} ag{279} ag{280} ag{281} ag{282} ag{283} ag{284} ag{285} ag{286} ag{287} ag{288} ag{289} ag{290} ag{291} ag{292} ag{293} ag{294} ag{295} ag{296} ag{297} ag{298} ag{299} ag{300} ag{301} ag{302} ag{303} ag{304} ag{305} ag{306} ag{307} ag{308} ag{309} ag{310} ag{311} ag{312} ag{313} ag{314} ag{315} ag{316} ag{317} ag{318} ag{319} ag{320} ag{321} ag{322} ag{323} ag{324} ag{325} ag{326} ag{327} ag{328} ag{329} ag{330} ag{331} ag{332} ag{333} ag{334} ag{335} ag{336} ag{337} ag{338} ag{339} ag{340} ag{341} ag{342} ag{343} ag{344} ag{345} ag{346} ag{347} ag{348} ag{349} ag{350} ag{351} ag{352} ag{353} ag{354} ag{355} ag{356} ag{357} ag{358} ag{359} ag{360} ag{361} ag{362} ag{363} ag{364} ag{365} ag{366} ag{367} ag{368} ag{369} ag{370} ag{371} ag{372} ag{373} ag{374} ag{375} ag{376} ag{377} ag{378} ag{379} ag{380} ag{381} ag{382} ag{383} ag{384} ag{385} ag{386} ag{387} ag{388} ag{389} ag{390} ag{391} ag{392} ag{393} ag{394} ag{395} ag{396} ag{397} ag{398} ag{399} ag{400} ag{401} ag{402} ag{403} ag{404} ag{405} ag{406} ag{407} ag{408} ag{409} ag{410} ag{411} ag{412} ag{413} ag{414} ag{415} ag{416} ag{417} ag{418} ag{419} ag{420} ag{421} ag{422} ag{423} ag{424} ag{425} ag{426} ag{427} ag{428} ag{429} ag{430} ag{431} ag{432} ag{433} ag{434} ag{435} ag{436} ag{437} ag{438} ag{439} ag{440} ag{441} ag{442} ag{443} ag{444} ag{445} ag{446} ag{447} ag{448} ag{449} ag{450} ag{451} ag{452} ag{453} ag{454} ag{455} ag{456} ag{457} ag{458} ag{459} ag{460} ag{461} ag{462} ag{463} ag{464} ag{465} ag{466} ag{467} ag{468} ag{469} ag{470} ag{471} ag{472} ag{473} ag{474} ag{475} ag{476} ag{477} ag{478} ag{479} ag{480} ag{481} ag{482} ag{483} ag{484} ag{485} ag{486} ag{487} ag{488} ag{489} ag{490} ag{491} ag{492} ag{493} ag{494} ag{495} ag{496} ag{497} ag{498} ag{499} ag{500} ag{501} ag{502} ag{503} ag{504} ag{505} ag{506} ag{507} ag{508} ag{509} ag{510} ag{511} ag{512} ag{513} ag{514} ag{515} ag{516} ag{517} ag{518} ag{519} ag{520} ag{521} ag{522} ag{523} ag{524} ag{525} ag{526} ag{527} ag{528} ag{529} ag{530} ag{531} ag{532} ag{533} ag{534} ag{535} ag{536} ag{537} ag{538} ag{539} ag{540} ag{541} ag{542} ag{543} ag{544} ag{545} ag{546} ag{547} ag{548} ag{549} ag{550} ag{551} ag{552} ag{553} ag{554} ag{555} ag{556} ag{557} ag{558} ag{559} ag{560} ag{561} \ ag{562} ag{563} ag{564} ag{565}\ ag{566} ag{567} ag{568} ag{569} ag{570} ag{571} ag{572} ag{573} ag{574} ag{575} ag{576} ag{577} ag{578} ag{579} ag{580} ag{581} ag{582} ag{583} ag{584} ag{585} ag{586} ag{587} ag{588} ag{589} ag{590} ag{591} ag{592} ag{593} ag{594} ag{595} ag{596} ag{597} ag{598} ag{599} ag{600} ag{601} ag{602} ag{603} ag{604} ag{605} ag{606} ag{607} ag{608} ag{609} ag{610} ag{611} ag{612} ag{613}\ ag{614} ag{615} ag{616} ag{617} ag{618} ag{619} ag{620} ag{621} ag{622} ag{623} ag{624} ag{625} ag{626} ag{627} ag{628} ag{629} ag{630} ag{631} ag{632} ag{633} ag{634} ag{635} ag{636} ag{637} ag{638} ag{639} ag{640} ag{641} ag{642} ag{643} ag{644} ag{645} ag{646} ag{647} ag{648} ag{649} ag{650} ag{651} ag{652} ag{653} ag{654} ag{655} ag{656} ag{657} ag{658} ag{659} ag{660} ag{661} ag{662} ag{663} ag{664} ag{665} ag{666} ag{667} ag{668} ag{669} ag{670} ag{671} ag{672} ag{673} ag{674} ag{675} ag{676} ag{677} ag{678} ag{679} ag{680} ag{681} ag{682} ag{683} ag{684} ag{685} ag{686} ag{687} ag{688} ag{689} ag{690} ag{691} ag{692}\ ag{693} ag{694}}
otag extbf{F}{ ext{net}} = m extbf{a} ag{26} ag{27} ag{28} ag{29} ag{30} ag{31} ag{32} ag{33} ag{34} ag{35} ag{36} ag{37} ag{38} ag{39} ag{40} ag{41} ag{42} ag{43} ag{44} ag{45} ag{46} ag{47} ag{48} ag{49} ag{50} ag{51} ag{52} ag{53} ag{54} ag{55} ag{56} ag{57} ag{58} ag{59} ag{60} ag{61} ag{62} ag{63} ag{64} ag{65} ag{66} ag{67} ag{68} ag{69} ag{70} ag{71} ag{72} ag{73} ag{74} ag{75} ag{76} ag{77} ag{78} ag{79} ag{80} ag{81} ag{82} ag{83} ag{84} ag{85} ag{86} ag{87} ag{88} ag{89} ag{90} ag{91} ag{92} ag{93} ag{94} ag{95} ag{96} ag{97} ag{98} ag{99} ag{100} ag{101} ag{102} ag{103} ag{104} ag{105} ag{106} ag{107} ag{108} ag{109} ag{110} ag{111} ag{112} ag{113} ag{114} ag{115} ag{116} ag{117} ag{118} ag{119} ag{120} ag{121} ag{122} ag{123} ag{124} ag{125} ag{126} ag{127} ag{128} ag{129} ag{130} ag{131} ag{132} ag{133} ag{134} ag{135} ag{136} ag{137} ag{138} ag{139} ag{140} ag{141} ag{142} ag{143} ag{144} ag{145} ag{146}} \
otagoldsymbol{F}{ ext{net},x} = moldsymbol{a}x\
otagoldsymbol{F}{ ext{net},y} = moldsymbol{a}y\
otagoldsymbol{F}{ ext{net},z} = moldsymbol{a}z ag{147} ag{148} ag{149} ag{150} ag{151} ag{152} ag{153} ag{154} ag{155} ag{156} ag{157} ag{158} ag{159} ag{160} ag{161} ag{162} ag{163} ag{164} ag{165} ag{166} ag{167} ag{168} ag{169} ag{170} ag{171} ag{172} ag{173} ag{174} ag{175} ag{176} ag{177} ag{178} ag{179} ag{180} ag{181} ag{182} ag{183} ag{184} ag{185} ag{186} ag{187} ag{188} ag{189} ag{190} ag{191} ag{192} ag{193} ag{194} ag{195} ag{196} ag{197} ag{198} ag{199} ag{200} ag{201} ag{202} ag{203} ag{204} ag{205} ag{206} ag{207} ag{208} ag{209} ag{210} ag{211} ag{212} ag{213} ag{214} ag{215} ag{216} ag{217} ag{218} ag{219} ag{220} ag{221} ag{222} ag{223} ag{224} ag{225} ag{226} ag{227} ag{228} ag{229} ag{230} ag{231} ag{232} ag{233} ag{234} ag{235} ag{236} ag{237} ag{238} ag{239} ag{240} ag{241} ag{242} ag{243} ag{244} ag{245} ag{246} ag{247} ag{248} ag{249} ag{250} ag{251} ag{252} ag{253} ag{254} ag{255} ag{256} ag{257} ag{258} ag{259} ag{260} ag{261} ag{262} ag{263} ag{264} ag{265} ag{266} ag{267} ag{268} ag{269} ag{270} ag{271} ag{272} ag{273} ag{274}
otag extbf{F}{ ext{net},x} = m extbf{a}x\
otag extbf{F}{ ext{net},y} = m extbf{a}y\
otag extbf{F}{ ext{net},z} = m extbf{a}z ag{275} ag{276} ag{277} ag{278} ag{279} ag{280} ag{281} ag{282} ag{283} ag{284} ag{285} ag{286} ag{287} ag{288} ag{289} ag{290} ag{291} ag{292} ag{293} ag{294} ag{295} ag{296} ag{297} ag{298} ag{299} ag{300} ag{301} ag{302} ag{303} ag{304} ag{305} ag{306} ag{307} ag{308} ag{309} ag{310} ag{311} ag{312} ag{313} ag{314} ag{315} ag{316} ag{317} ag{318} ag{319} ag{320} ag{321} ag{322} ag{323} ag{324} ag{325} ag{326} ag{327} ag{328} ag{329} ag{330} ag{331} ag{332} ag{333} ag{334} ag{335} ag{336} ag{337} ag{338} ag{339} ag{340} ag{341} ag{342} ag{343} ag{344} ag{345} ag{346} ag{347} ag{348} ag{349} ag{350} ag{351} ag{352} ag{353} ag{354} ag{355} ag{356} ag{357} ag{358} ag{359} ag{360} ag{361} ag{362} ag{363} ag{364} ag{365} ag{366} ag{367} ag{368} ag{369} ag{370} ag{371} ag{372} ag{373} ag{374} ag{375} ag{376} ag{377} ag{378} ag{379} ag{380} ag{381} ag{382} ag{383} ag{384} ag{385} ag{386} ag{387} ag{388} ag{389} ag{390} ag{391} ag{392} ag{393} ag{394} ag{395} ag{396} ag{397} ag{398} ag{399} ag{400} ag{401} ag{402} ag{403} ag{404} ag{405} ag{406} ag{407} ag{408} ag{409} ag{410} ag{411} ag{412} ag{413} ag{414} ag{415} ag{416} ag{417}
otag extbf{F}{ ext{net},x} = m extbf{a}x\
otag extbf{F}{ ext{net},y} = m extbf{a}y\
otag extbf{F}{ ext{net},z} = m extbf{a}z ag{418} ag{419} ag{420} ag{421} ag{422} ag{423} ag{424} ag{425} ag{426} ag{427} ag{428} ag{429} ag{430} ag{431} ag{432} ag{433} ag{434} ag{435} ag{436} ag{437} ag{438} ag{439} ag{440} ag{441} ag{442} ag{443} ag{444} ag{445} ag{446} ag{447} ag{448} ag{449} ag{450} ag{451} ag{452} ag{453} ag{454} ag{455} ag{456} ag{457} ag{458} ag{459} ag{460} ag{461} ag{462} ag{463} ag{464} ag{465} ag{466} ag{467} ag{468} ag{469} ag{470} ag{471} ag{472} ag{473} ag{474} ag{475} ag{476} ag{477} ag{478} ag{479} ag{480} ag{481} ag{482} ag{483} ag{484} ag{485} ag{486} ag{487} ag{488} ag{489} ag{490} ag{491} ag{492} ag{493} ag{494} ag{495} ag{496} ag{497} ag{498} ag{499} ag{500} ag{501} ag{502} ag{503} ag{504} ag{505} ag{506} ag{507} ag{508} ag{509} ag{510} ag{511} ag{512} ag{513} ag{514} ag{515}
otag extbf{F}{ ext{net},x} = m extbf{a}x\
otag extbf{F}{ ext{net},y} = m extbf{a}y\
otag extbf{F}{ ext{net},z} = m extbf{a}z ag{516} ag{517} ag{518} ag{519} ag{520}
otag\
otag extbf{F}{ ext{net},x} = m extbf{a}x \
otag extbf{F}{ ext{net},y} = m extbf{a}y \
otag extbf{F}{ ext{net},z} = m extbf{a}z
- Vector Equation: oldsymbol{ ext{F}}_{ ext{net}} = moldsymbol{a} is a vector equation, meaning it holds true for each component independently.
- Inertia: Can be thought of as resistance to changes in motion. The mass (m) in Newton's second law is sometimes called inertial mass.
- Inertial Mass vs. Gravitational Mass: The mass in oldsymbol{ ext{F}}_{ ext{net}} = moldsymbol{a} (inertial mass) is experimentally found to be exactly equivalent to the mass that dictates gravitational force (gravitational mass), though there's no fundamental reason it must be so. This equivalence is a profound principle in physics.
- First Law as a Special Case: If oldsymbol{ ext{F}}_{ ext{net}} = 0, then oldsymbol{a} = 0, meaning velocity is constant (magnitude and direction), which is the statement of the First Law.
Newton's Third Law (Action-Reaction Pairs)
- Formal Statement: If object 1 exerts a force (oldsymbol{ ext{F}}{12}) on object 2, then object 2 simultaneously exerts an equal and opposite force (oldsymbol{ ext{F}}{21}) on object 1.
- Mathematically: oldsymbol{ ext{F}}{12} = -oldsymbol{ ext{F}}{21}
- Terminology: Often called