trnsltionl nd rottionl nlogues trnsltionl ( liner ) motion rottionl motion trnsltionl displcement d A = r A Δθ ngulr displcement Δx Δy Must use rdins. Δθ (delt thet) unit = m unit = rd trnsltionl elocity A = r A ω ngulr elocity rolling: cm = r cm ω ω (omeg) unit = m/s unit = rd/s trnsltionl ccelertion unit = m/s Must use rdins At = ±r A α Must use rdins. ngulr ccelertion α (lph) unit = rd/s mss m orce F unit = kg unit = N moment o inerti I = mr point msses only unit = kg m r is the distnce beteen the point mss nd the the xis o rottion. To ind I or n extended object use tble. torque τ (tu) unit = N m Newton s Second Lw or trnsltion F x = m cmx F y = m cmy trnsltionl kinetic energy trk = 1 m cm unit = J τ = F r = F r Newton s Second Lw or rottion τ = Iα rottionl kinetic energy 1 rotk = Iω unit = J
the kinemtics ribles trnsltionl motion Δ x t Δ y ix x x iy y y t rottionl motion Δ θ ω ω α t i the constnt-ccelertion kinemtics equtions trnsltionl x-equtions missing ribles rottionl equtions missing ribles = t Δ x ω = ω + αt Δ θ x ix + x ix + x x ωi + ω Δ x = t Δ θ = t = + Δx t ω = ω + αδ θ x ix x i t 1 Δ x = ix t + xt 1 x Δ θ = ω i t + αt ω 1 Δ x = xt xt 1 ix Δ θ = ω t αt ω i You he to use consistent units in kinemtics eqution but you do not he to use SI units. systemtic method or soling constnt-ccelertion rottionl kinemtics problems 1. Drw the object s pth. Lbel the initil nd inl positions. Drw the directions o ω nd α clockwise or counterclockwise.. I you hen t done so lredy write down positie direction CW or CCW. It is usully best to choose the direction o motion s the positie direction. 3. Write down ll o the kinemtics ribles. Underneth the ribles write down the gien lues including signs nd indicte the question with?. 4. When you know lues or three o the kinemtics ribles you cn choose n eqution. Identiy the one rible you don t cre bout nd pick the eqution tht is missing tht rible. Plug in nd sole. Write your inl nswer with sign nd units. i α
How to ind the moment o inerti I o mss The moment o inerti indictes the object s rottionl inerti i.e. how hrd it is to chnge the rottion o the object. The moment o inerti o collection o objects is the sum o the indiidul moments o inerti. point mss method When you re not gien the object s dimensions or shpe. Drw r rom the xis o rottion to the loction o the mss. Determine r. I the mss is locted on xis o rottion then r =0 so I=0. Determine I = mr where m is the mss Drw the xis o rottion or piot point extended object method When you re gien the object s dimensions or shpe. Wht is the object s shpe? Is the object hollow or solid? Where is the xis o rottion? Find the prt o the Rottionl Inertis tble tht mtches these three chrcteristics o the object. I nothing in the tble hs the right xis o rottion use the tble to ind I cm the rottionl inerti bout n xis through the center o mss. Then i the ctul xis is prllel to the center-o-mss xis you cn use the prllel-xis theorem to ind I round the ctul xis o rottion: I = I cm + Md where M is the mss nd d is the perpendiculr distnce between the center-o-mss xis nd the ctul xis. As cn be seen rom the ormuls or I the moment o inerti hs units o kg m.
How to ind the torque exerted by n indiidul orce: two methods The torque indictes how eectie the orce is t chnging the object s rottion. r method usully best when you know the ngle between F nd r r method usully best when you don t. know the ngle between F nd r. 1. Drw the xis o rottion or piot point. Drw F t its point o ppliction. Determine F in newtons. 3. Drw r rom the xis o rottion to the point o ppliction o F. Determine r in meters. 3. Drw the line o orce line running through the point o ppliction o the orce nd prllel to F. I the orce is being pplied directly to the xis o rottion then r =0 so τ=0. (A orce pplied directly to the xis o rottion cnnot ect rottion.) 4. Locte nd determine θ. θ is the ngle between F nd r. Be creul: Just becuse you re gien n ngle in the problem doesn t men tht tht ngle is θ! 4. Drw r rom the xis o rottion perpendiculr to the line o orce. Determine r in meters. ( r is lso clled the leer rm. ) I the orce is being pplied directly to the xis o rottion then r =0 so τ=0. (A orce pplied directly to the xis o rottion cnnot ect rottion.) I the line o orce runs through the xis o rottion then r =0 so τ=0. (A orce tht is prllel to r cnnot ect rottion.) 5. Choose positie direction or torque either clockwise or counterclockwise. I the object is rotting it is best to choose the direction o rottion s the positie direction. I there is more thn one torque you need to use the sme positie direction or ll o them. 6. Determine the sign o the torque by sking whether the orce would mke r rotte clockwise or counterclockwise i it were pplying the only torque on r. 7. Determine τ = ± r F sinθ. By using the term sin θ we re sying tht only the component o the orce tht is perpendiculr to r cn exert torque. As cn be seen rom the ormuls or τ torque hs units o You must use S.I. units in the ormuls or torque in step 7. 6. Determine the sign o the torque by sking whether the orce would mke r rotte clockwise or counterclockwise i it were pplying the only torque on r. 7. Determine τ = ± r F. N m.
How to use Newton s Second Lw or rottionl motion 1. Identiy ll the objects. Usully ech thing or which you re gien mss or moment o inerti is treted s seprte object.. For ech object identiy ll the orces on the object nd where they re being pplied. 3. Identiy the xis o rottion or piot point. 4. Choose the directions o motion s the positie directions or the x xis the y xis nd or rottion (clockwise or counterclockwise). 5. Identiy the x nd y components o ech orce including the signs. Identiy the torque rom ech orce including the signs. Orgnize this inormtion into tble o components nd torques. 6. Identiy the moment o inerti I or ny object undergoing rottionl motion. I the object hs multiple prts identiy the I or ech indiidul prt nd then dd them up to ind the totl I. To ind the I o point mss use I=mr. For n extended object use Rottionl Inertis tble to ind the I. 7. Write down the pproprite ersions o Newton s Second Lw or ech object. F x = m cmx F y = m cmy τ = Iα Plug the pproprite inormtion into ech eqution. 8. I necessry use At = ±r A α to substitute or cmt or α. 9. When you he s mny equtions s unknowns reduce the number o ribles by soling one o the equtions or rible nd substituting or tht rible into the remining equtions; repet s mny times s necessry.