Miss Millie Millie

Newton’s Laws of Motion Principia

1. Everybodycontinuesinitsstateofrest,oruniformmotioninastraightline,unlesscompelledtochangethatstatebyforcesacteduponit.

2. Thechangeofmotion(momentum)ofanobjectisproportionaltotheforceimpresseduponit,andismadeinthedirectioninwhichthefibreisimpressed.

3. Toeveryactionthereisalwaysanopposedorequalreaction.

Galilean transformations

𝑥 = 𝑥! + 𝑣𝑡 𝑦 = 𝑦′𝑧 = 𝑧′

!"

!"= !"!

!"+ 𝑣 𝑎 = 𝑎′

Whatifthesecondframeacceleratesw.r.tthefirst?

𝑎! = 𝑎 −𝑑𝑢𝑑𝑡

𝑧’

𝑦’

𝑥’

𝑧

𝑦

𝑥

𝑃

�̃�

�̃�′

𝑢!𝑡

𝑢

conservative forces ~Areindependentofpath~

𝑓! = −∇ 𝑈

! 𝑓!!!

!!∙ 𝑑�̃� = −! ∇ 𝑈

!

! 𝑑�̃� = [𝑈]!

!

= 𝑈(𝑄) − 𝑈(𝑃)

∴ ∆𝐾 = −∆𝑈

𝐸 = 𝐾 + 𝑈 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

E

U

KU x

y

Dot Product

𝑎! ∙ 𝑏! = |𝑎!|!𝑏!! cos 𝜃

𝑖𝑓 𝑎! ∙ 𝑏! = 0 𝑎! ⊥ 𝑏!

𝑎! = 0 & 𝑏! = 0

𝑎! ∙ 𝑎! = 𝑎!!

𝜃

𝑎!

𝑏!

3rd Law

𝑓!,! = −𝑓!,!~aparticlecannotexertaforceonitself~

𝑓! = −𝑓! 𝑜𝑛𝑙𝑦 𝑖𝑓 𝑓! = 0

𝑚! 𝑚!"

Conservation of linear momentum Inasystemofmutuallyinteractingobjects,fortheithobject:

𝑚!𝑑!𝑟!𝑑𝑡! = ! 𝑓!" + 𝑓!"# =→ 𝑎𝑙𝑙 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙 𝑓𝑜𝑟𝑐𝑒𝑠 𝑐𝑎𝑛𝑐𝑒𝑙 𝑎𝑛𝑑 𝑓!"# = 0

AngularMomentum

𝑃 = !𝑚!𝑑𝑟!𝑑𝑡 → 𝑡𝑜𝑡𝑎𝑙 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚

𝑙! = 𝑚!𝑟!𝑑𝑟!𝑑𝑡

∴ 𝐿 = ! 𝑙! → 𝑇𝑜𝑡𝑎𝑙 𝑀𝑜𝑚𝑒𝑛𝑡𝑢𝑚 = 0 (𝑢𝑛𝑙𝑒𝑠𝑠 𝑡ℎ𝑒𝑟𝑒 𝑖𝑠 𝑓!"#)

2nd Law 𝑑𝑝𝑑𝑡 = 𝑓 = 𝑚

𝑑𝑣𝑑𝑡 = 𝑚

𝑑!𝑥𝑑𝑡

𝑣! ∙ 𝑓! = 𝑣! ∙ !𝑚𝑑𝑣𝑑𝑡! = 𝑚𝑣!

𝑑𝑣𝑑𝑡

=𝑚2

! 2𝑣!𝑑𝑣𝑑𝑡! =

𝑚2!𝑑𝑣𝑑𝑡𝑣! +

𝑑𝑣𝑑𝑡𝑣!! =

𝑚2𝑑𝑑𝑡

(𝑣! ∙ 𝑣!) =𝑑𝑑𝑡𝑚𝑣!

2=𝑑𝐾𝑑𝑡

∆𝐾 = ! 𝑓 ! ∙ 𝑣!!!

!! 𝑑𝑡 = ! 𝑓!

!

!∙ 𝑑�̃� =𝑊

Non-isolated systems Inasystemofmutuallyinteractingobjects,withanexternalforce,fortheithobject:

𝑚!𝑑!𝑟!𝑑𝑡! = ! 𝑓!" + 𝑓!"#

𝑑𝑃𝑑𝑡 = 𝑓!"# = ! 𝑓!"#(!)

Totalsystemmomentumdependson𝑓!"# Torque

𝑑𝐿𝑑𝑡 = 𝜏 = ! 𝑟! × 𝑓!"#(!)