Phscs 45 Fomula Sheet Eam 3 One-dmensonal knematcs Vectos dsplacement: Δ total dstance taveled aveage speed total tme Δ aveage veloct: vav t t Δ nstantaneous veloct: v lm Δ t v aveage acceleaton: aav t d v t dv d nstantaneous acceleaton: a lm One-dmensonal moton wth constant acceleaton: () v v + at () + ( v + v)t (3) + v t + at v v + a (4) ( ) Fee all (postve decton o taken to be upwad) and a g n the above 4 equatons o knematcs: () v v gt () + ( v + v)t (3) + v t gt v v g (4) ( ) Vectos n -D I a vecto A s wtten n component om as A A ˆ + A ˆ A, A then: Gettng magntude and decton o A om the components: A A + A (magntude o A ) A θ actan (decton o A ) A
Fomula Sheet Eam 3 Page Gettng components om magntude and decton: A A cosθ ( component o A ) A snθ ( component o A θ undestood to be the angle that ) } A makes wth the postve as A Vectos n 3-D I A A ˆ + A ˆ + A ˆ A, A, A then: A A + A + A -D Knematcs poston vecto: () t () t ˆ + ( t) ˆ ( t), ( t) Δ Δ Δ aveage veloct: v av, d d d nstantaneous veloct: v, aveage acceleaton: aav, nstantaneous acceleaton: dv dv dv a, d d d a, nstantaneous speed v (magntude o the nstantaneous veloct) 3-D knematcs poston vecto: () t () t ˆ + ( t) ˆ + ( t) ˆ ( t), ( t), ( t) Δ Δ Δ Δ aveage veloct: v av,, d d d d nstantaneous veloct: v,, aveage acceleaton: aav,, nstantaneous acceleaton: dv dv dv dv a,, d d d d a,, nstantaneous speed v (magntude o the nstantaneous veloct)
Fomula Sheet Eam 3 Page 3 Pojectle Moton decton (moton wth constant veloct): a v v + v t decton (ee all... postve decton o taken to be upwad): () v v gt () + ( v + v) t (3) + vt gt v v g Relatve Moton (4) ( ) vpa vpb + vba Newton s Laws o Moton Two boad categoes o oces: contact oces (objects n contact wth one anothe) and eld oces (objects not n contact wth one anothe). Gavt s the onl eld oce we wll deal wth n ths couse. Weght: w mg st law: v s constant unless object (o sstem) epeences net etenal oce. nd law : F ma mples thee statements, n geneal: F ma, F ma, and F ma *o sstems o objects, F m a et ss ss 3 d law: Wheneve one object eets a oce on a second object, the second eets a oce on the st; these two oces ae equal n magntude and opposte n decton: F F Equlbum An object s sad to be n equlbum (eall n tanslatonal equlbum) : F a *eall mples thee sepaate equements o tanslatonal equlbum: ) F a ) F a 3) F a Fcton oces s μs k μk n n
Fomula Sheet Eam 3 Page 4 Ccula Moton nd law (centpetal decton): ( F ) maad Radal (centpetal) acceleaton: a ad ad v a I thee s a tangental acceleaton a also, then: dv a Dot Poduct (Scala Poduct) Fo an two vectos A A, A, A A B AB cosθ, n whch: and B B, B, B : A A + A + A Wok B B + B + B Altenatve (equvalent) denton o dot poduct: A B A B + A B + A B Wok Vaable Foces: W F d (geneal denton o wok) I F has onl an component and ths component depends on, then: W F( ) d Constant Foces: W F Δ (-D o 3-D path) W FΔ (-D path) Spngs Hooke s law: F k. k the spng constant o the oce constant. Potental eneg stoed n a spng (elastc potental eneg): Wok-Eneg Theoem ΔK W net Uel k Knetc eneg: K mv Powe Instantaneous Powe:
Fomula Sheet Eam 3 Page 5 de Geneal Denton: Rate at whch eneg beng suppled b o to a sstem: P I eneg comes om wok beng done, then the powe s the ate at whch wok s done: dw P P F v (altenatve denton) Aveage Powe: ΔE ΔW Pav Consevatve Foces, Potental Eneg, and Consevaton o Eneg Consevatve Foces: Wok done b a consevatve oce: W c ΔU, o some potental eneg U Wok done s ndependent o path. Wok done gong once aound closed path s eo. Potental Eneg: Gavtatonal potental eneg (nea suace o Eath): U gav mg Elastc Potental Eneg: Uel Total mechancal eneg: E K + U Nonconsevatve Foces: Wnc Δ E I W nc, E conseved. Foce and Potental Eneg F F du d du d du F d Lnea Momentum Lnea momentum: k p mv p mv p mv p mv dp Δp Fnet Fnet av Δ t Newton s second law: ( ) net F s net (etenal) oce ( Fnet F ) Impulse Vang oce: t J F t Constant oce: J F (geneal denton o mpulse)
Fomula Sheet Eam 3 Page 6 Impulse-momentum theoem: Jnet Δp, n whch: Collsons t Jnet Fnet (vang oce) o Jnet Fnet (constant oce) t Two boad categoes: head-on and glancng Fo each catego, thee classes:. elastc: p and K conseved. nelastc: p conseved, K not 3. completel nelastc: p conseved, K not, objects stck togethe Head-on collsons:. elastc: v + m v m v m v (p-consevaton) m + v v v + v + ( othe eq deved n class). nelastc: m v + mv mv + mv (p-consevaton) 3. completel nelastc: v + m v m m v (p-consevaton) Glancng (-D) collsons: p p p p ( ) m + Cente o Mass and Sstems o Patcles m + m+ + mnn m + m+ + mnn X CM m+ m + + mn Mtot m + m + + mnn m + m + + mnn YCM m+ m + + mn Mtot nd law o sstem: dp ( F ), n whch P mv + mv+ + mnvn s total momentum o sstem. et P MtotvCM I mass o sstem s constant, then: ( F ) MtotaCM Rotatonal Knematcs et s angle θ n adans: θ (s length o ac swept out) angula dsplacement: Δ θ θ θ Δθ θ θ aveage angula veloct: ω av t t dθ nstantaneous angula veloct: ω Δω ω aveage angula acceleaton: α av Δ t t ω t
Fomula Sheet Eam 3 Page 7 nstantaneous angula acceleaton: dω d θ α *θ undestood to be n adans n all o the above Rotatonal moton wth constant angula acceleaton: () ω ω + α t () θ θ + ( ω + ω )t (3) θ θ + ω t + α t ω ω + α θ θ (4) ( ) Rotatonal and Lnea Quanttes tan tan v v ω a a α Radal (centpetal) acceleaton: a a ω ad Rollng Wthout Slppng vcm Rω ( v tanslatonal speed o cente o mass o ollng object) Rotatonal Knetc Eneg and Moment o Ineta Geneal Fomula o Moment o Ineta: I dm Moment o Ineta o Collecton o N Pont Masses: m Moments o Ineta o Dstbuted Objects: N I Rotatonal Knetc Eneg: K ot Iω Total Knetc Eneg: K total K tans + K ot MvCM + I CM ω
Fomula Sheet Eam 3 Page 8 (Note: Hee v CM s the tanslatonal veloct o the cente o mass and I CM s the moment o neta about the cente o mass.) Paallel-as Theoem I I + Md P CM Pependcula-as Theoem I I + I Coss Poduct (Vecto Poduct) A B ( A ) ˆ ( ) ˆ B AB + AB AB + ( AB AB) ˆ A B ABsnφ Toque and Angula Acceleaton Toque: τ F τ F Relaton Between Toque and Angula Acceleaton (Newton s Second Law o Rotatonal Moton): τ I α Angula Momentum net Fo pont patcle: L p. Fo gd bod otatng about a ed smmet as: L Iω. dl Newton s Second Law o Rotatonal Moton: τ net. (Note hee that τ net s the net etenal toque.) Equlbum Thee condtons equed o tue equlbum (tanslatonal and otatonal equlbum):. F (.e., a ). F (.e., a ) 3. τ o an as (.e., α about an as) net Gavt mm Fgav G (Newton s law o unvesal gavtaton) N m G 6.67 (gavtatonal constant) kg g GM (acceleaton due to gavt above planet, moon, etc., o mass M)
Fomula Sheet Eam 3 Page 9 mm U G (gavtatonal potental eneg o an two masses m and m ) gav GM vesc (escape speed) R Smple Hamonc Moton (SHM) Relatonshps among equenc, angula equenc and peod o snusods: T ω π π T ω Mass on Spng Acos ωt+ φ (most geneal om o dsplacement as a uncton o tme) ( ) ω k m m T π k v A + ω v φ tan ω Smple Pendulum L T π g Phscal Pendulum T π I mgd Sola Sstem Data adus o Eath: adus o Moon: mass o Eath: mass o Moon: mass o Sun: R 6.37 m E R.74 m M 6 6 4 M 5.97 kg E 7.35 kg 3. kg Eath-Moon dstance (cente-to-cente): 3.84 m Eath-Sun dstance (cente-to-cente):.5 m 8