m m m00 kg dult, m0 kg bby. he seesw stts fom est. Which diection will it ottes? ( Counte-Clockwise (b Clockwise ( (c o ottion ti (d ot enough infomtion Effect of Constnt et oque.3 A constnt non-zeo toque is exeted on wheel. Which of the following quntities must be chnging?. ngul position. ngul velocity 3. ngul cceletion 4. moment of ineti 5. kiic enegy 6. the mss cente loction A.,, 3 B. 4, 5, 6 C.,, 5 D.,, 3, 4 E., 3, 5
Exmple: second lw fo ottion PP060-50: A toque of 3.0 m on cetin wheel cuses n ngul cceletion of 5.0 d/s. Wht is the wheel's ottionl ineti? Second w exmple: fo n unblnced b B is mssless nd oiginlly hoizontl Rottion xis t fulcum point hs zeo toque +y Find ngul cceletion of b nd the line cceletion of m just fte you let go Use: I o,i + I whee: I I + I m + toque l I bout pivot m Wht hppened to sin( in moment m? m m g + m Wht if b is not hoizontl? m g fulcum Constints: Using specific numbes: et m m m 0 cm, 80 cm g g m g g(0. - 0.8 + 0. + 0.8 8.65 d/s - +.7 m/s Clockwise Acceletes UP
See Sw 3.. Suppose eveything is s it ws in the peceding exmple, but the b is O hoizontl. Assume both msses e equl. Which of the following is the coect eqution fo the ngul cceletion? A B ( g + ( g + cos( m g fulcum m g C D E ( gsin( + ( gcos( + ( g + sin( I See Sw - Solution 3.. Suppose eveything is s it ws in the peceding exmple, but the b is O hoizontl. Assume both msses e equl. Which of the following is the coect eqution fo the ngul cceletion? / I I m g o,i + cos( cos( fulcum m g I I + I m + [m m m ] g cos( + m m Fo equl msses m m m ( If b is blnced when it s gcos( hoizontl, does tht chnge + when it is not hoizontl? 3
Method fo solving (complex poblems using the Second w Mny components in the system mens sevel ( unknowns. need n equl numbe of independent equtions Method: Dw o sketch system. Adopt coodintes, nme the vibles, indicte ottion xes, list the known nd unknown quntities, Dw fee body digms of key pts. Show foces t thei points of ppliction. find toques bout (common xis ote: cn hve My need to pply Second w twice to ech pt nsltion: F F F.eq. 0 but i m.ne. 0 Rottion: i I Mke sue thee e enough ( equtions; thee my be constint equtions (ext conditions connecting unknowns Simplify nd solve the set of (simultneous equtions. Intepet the finl fomuls. Do they mke intuitive sense? Refe bck to the sketches nd oiginl poblem Clculte numeicl esults, nd snity check nwes (e.g., ight ode of nitude? Exmple: oque nd Angul Acceletion of Wheel Anlysis ppoch: Bek into two sub-systems wheel is cceleted ngully by tension block is cceleted linely by weight, with tension opposing Fee body digms shown he wheel is otting nd so we pply Σ Ι he tension supplies the toque vi tngentil foce he mss is moving in stight line, so pply ewton s Second w ΣF y m y - How to connect the two poblems bove? eed constint linking line cceletion to 4
Appliction of Method using the Second w Cod wpped ound disk, hnging weight Cod does not slip o stetch constint Disk s ottionl ineti slows cceletions et m. kg, M.5 kg, 0. m Fo Find mss cceletion m: of mss m, find fo disk F m y y m (g Unknowns:, suppot foce t xis O hs FBD fo disk, with xis t o : zeo toque Mg 0 + I I M m(g I M Unknowns:, fom no So f: Equtions, 3 unknowns eed constint: Substitute nd solve: + slipping ssumption m - ( 4 d/s popotionl M M (m + M/ to g m ( + M M (m + M/ ( 4.8 m/s Exmple: Hevy pulley Atwood s mchine ppliction of ewton s nd w Given the numeicl vlues: m 5.0 kg, m 5.5 kg 0. m, M60kg 6.0 I 0.0 kg-m O M, I Find the cceletions of the hnging weights. Ae they up o down? b Find the ngul cceletion of the pulley. Is it CW o CCW? c Find the tensions in ech cod. d How long does ittke the 55k 5.5 kg weight htto fll 0.6 m fom est? m m Sttegy: Apply ewton s Second w (line nd ottionl fom to ll thee bodies, connected by constints. Mke sue thee e s mny independent equtions s thee e unknowns 5
Hevy pulley Atwood s mchine: solution Apply Second w: (I F F m i (ech component (II I i (pulley If I 0 poblem simplifies Fo m use I: m -. m ( + g m g positive negtive (down Fo m use I gin: m choose y positive up fo - m g both msses. m ( + g CCW ottion positive m g positive positive (up Fee fll if o 0 Constints: cod cnnot stetch o slip + only of these e independent y O M, I Mg y m m ote: if I 0, then. Result using & becomes: m m - m + m g Atwood s solution, continued Fo the Pulley use II: + I i ( 3. I positive > Check bility to solve: O M, I Mg 5 unknowns:,,,, Hve 5 independent equtions:,,3 nd constints Solution: Begin eliminting unknowns (tension equtions OE: m g m g OK! I i (m g + m - m - (m - m g + (m + m I - (m + m (m - m g I + (m + m (m - m g Result: intepettion is simple imiting cse: et I 0: mssless pulley (m m g I + (m + m (m m g (m m + et toque of hnging weights otl moment of Ineti m - m g m + m egtive if m > m : 6
Atwood s solution, numeicl evlution ( 5. 0 5. 5 ( 9.8 (0. (b -.58 d/s (CW 0.0 + ( 5. 0 5. 5 0. ( - + 0.3-03 0.3 m/s m/s (up (down 5.0 (9.8 + 0.3 5.5 (9.8 0.3 (c 506. 5. ( inceses tension ( deceses tension How long does m tke to fll 0.6 m fom est? d t d (d t.94 s 7