# total If no external forces act, the total linear momentum of the system is conserved. This occurs in collisions and explosions.

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1 Lesson 0: Collsons, Rotatonal netc Energy, Torque, Center o Graty (Sectons 7.8 Last te we used ewton s second law to deelop the pulse-oentu theore. In words, the theore states that the change n lnear oentu s caused by the pulse. In an equaton, p J F total t I no external orces act, the total lnear oentu o the syste s consered. Ths occurs n collsons and explosons. We dered the equatons or a one densonal elastc collson (knetc energy s consered) On any occasons the collson s not one densonal. What do we do then? Proble 8. An object o ass.0 kg (the projectle ) approaches a statonary object (the target ) at 8.0 /s. The projectle s delected through an angle o 90.0 and ts speed ater the collson s 6.0 /s. What s the speed o the target body ater the collson the collson s perectly elastc? Soluton: Snce the ncdent ass changes drecton ater the collson, we cannot use the equatons or an elastc one-densonal collson. Instead, we hae an elastc two-densonal collson. Intal Fnal Snce ths s a collson, there are no external orces actng and oentu s consered. Usng the dagra to take coponents: p p Lesson 0, page

2 Lesson 0: Collsons, Rotatonal netc Energy, Torque, Center o Graty (Sectons 7.8 Lesson 0, page x x p p sn sn 0 y y p p The unknowns are, and. There s a trck to elnate Square the two equatons sn sn and add the together sn sn Aboe we used the dentty sn Snce the collson s elastc, knetc energy s consered: Solng or and substtute nto the equaton ound when we squared and added the oentu equatons

3 Lesson 0: Collsons, Rotatonal netc Energy, Torque, Center o Graty (Sectons 7.8 Lesson 0, page 3.80/s 6.0 /s 8.0 /s 6.0 /s 8.0 /s 4 Exaple: Consder a two densonal elastc collson between two dentcal asses. One ass s ntally at rest. Soluton: Elastc collsons consere knetc energy Cancellng the ½ and the asses, snce =, The conseraton o lnear oentu condton s Agan, we can cancel the asses

4 Lesson 0: Collsons, Rotatonal netc Energy, Torque, Center o Graty (Sectons 7.8 The two boxed equatons can be nterpreted by the pcture Ater the collson the two asses hae eloctes perpendcular to each other. Chapter 8 Torque and Angular Moentu Reew o Chapter 5 We had a table coparng paraeters ro lnear and rotatonal oton. Today we ll n the table. Here t s Descrpton Lnear Rotatonal poston x dsplaceent x Rate o change o poston x x Aerage rate o change o poston x, a a t t x x l Instantaneous rate o change o poston t 0 t t 0 t x ax a, a Aerage rate o change o speed t t x ax l Instantaneous rate o change o speed t 0 t t 0 t Inerta I Inluence that causes acceleraton F Moentu p L Lesson 0, page 4

5 Lesson 0: Collsons, Rotatonal netc Energy, Torque, Center o Graty (Sectons 7.8 The relatons (oten physcal laws) or rotatonal oton can be ound by a sple substtuton o rotatonal arables or the correspondng lnear arables. Rotatonal netc energy A wheel suspended at ts axs can spn n space. Snce the ponts o the wheel are ong, the wheel has knetc energy. All the peces n a rgd body rean at the sae locaton relate to all the other peces. For a rotatng object, the parts urther away ro the axs o rotaton are ong aster. r The total knetc energy o all the peces wll be total total r r The quantty n parentheses s called the rotatonal nerta (or the oent o nerta) Fndng the Rotatonal Inerta (page 70). I the object conssts o a sall nuber o partcles, calculate the su I I n r n r n drectly.. For syetrcal objects wth sple geoetrc shapes, calculus can be used to peror the su. Table 8. (see below) lsts the results o these calculatons or the shapes ost coonly encountered. 3. Snce the rotatonal nerta s a su, you can always entally decopose the object nto seeral parts, nd the rotatonal nerta o each part, and then add the. Ths s an exaple o the dde-and-conquer proble-solng technque. Lesson 0, page 5

6 Lesson 0: Collsons, Rotatonal netc Energy, Torque, Center o Graty (Sectons 7.8 The rotatonal nerta depends on the locaton o the rotaton axs. The sae object wll hae a derent rotatonal nerta dependng on where t s rotatng. Look at the orula or a thn rod below. The rotatonal knetc energy o a rgd object rotatng wth angular elocty s Copare to the translatonal knetc energy I Torque A quantty related to orce, called torque, plays the role n rotaton that orce tsel plays n translaton. A torque s not separate ro a orce; t s possble to exert a torque wthout exertng a orce. Torque s a easure o how eecte a gen orce s at twstng or turnng soethng. The torque due to a orce depends o the agntude o the appled orce, the orce s pont o applcaton, and the orce s drecton. Lesson 0, page 6

7 Lesson 0: Collsons, Rotatonal netc Energy, Torque, Center o Graty (Sectons 7.8 Frst denton o torque rf r Because rotatons hae drectons, we assgn the + sgn to torques that cause counterclockwse rotatons, and sgn to torques that cause clockwse rotatons. What s the sgn o the torque n the gures aboe? Torques are easured n the unts o orce tes dstance. Ths s the sae densons as work. Howeer, torque has a derent eect than work. To keep the two concepts dstnct, we easure work n joules and torque n newton-eters. Second denton o torque r F r Fnd the leer ar (or oent ar) by extendng the lne o the orce and drawng a lne ro the axs o rotaton so that s crosses the lne o the orce at a rght angle. Fndng the leer ar s oten the ost dcult part o a torque proble. Lesson 0, page 7

8 Lesson 0: Collsons, Rotatonal netc Energy, Torque, Center o Graty (Sectons 7.8 Fndng Torques Usng the Leer Ar (p 78). Draw a lne parallel to the orce through the orce s pont o applcaton; ths lne s called the orce s lne o acton.. Draw a lne ro the rotaton axs to the lne o acton. Ths lne ust be perpendcular to both the axs and the lne o acton. The dstance ro the axs to the lne o acton along ths perpendcular lne s the leer ar (r ). I the lne o acton o the orce goes through the rotaton axs, the leer ar and the torque are both zero. 3. The agntude o the torque s the agntude o the orce tes the leer ar: r F 4. Deterne the algebrac sgn o the torque as beore. Center o Graty When the gratatonal orce acts on an object, all the sall peces o the object experence the gratatonal orce. Ths ery large nuber o orces taken around an axs wll create a torque. How do we deal wth that? Fortunately, we can greatly sply the proble. The total orce can be consdered to act a sngle pont called the center o graty. I the gratatonal eld s unor n agntude and drecton, the center o graty s located at the center o ass. becoes Work done by a torque Lesson 0, page 8

9 Lesson 0: Collsons, Rotatonal netc Energy, Torque, Center o Graty (Sectons 7.8 Lesson 0, page 9 The expresson or the work done s r r s F W Power s the rate o dong work t t W P

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