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1 Physcs 06 ecture 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap. to 3 Rotatonal quanttes as vectors Cross product Torque epressed as a vector Angular momentum defned Angular momentum as a vector Newton s second law n vector form So far: smple (planar) geometres Rotatonal quanttes Δ,, α, τ, etc represented by scalars Rotaton as was specfed smply as CCW or CW Problems were dmensonal wth a perpendcular rotaton as Now: 3D geometres Cross product represents rotatonal quanttes as vectors: Cross products pont along nstantaneous aes of rotaton Drectons of rotaton aes can be calculated and summed up lke other vectors, e.g., τ r F τ rf sn( ) τ τ all Angular momentum - new rotatonal quantty. ke lnear momentum, t s conserved for solated systems Defnton: l r p p mv l rp sn( ) tot l all Second aw n terms of conserved quanttes: near: F dp ma F p s constant f F d 0 Rotatonal: τ Iα τ s constant f τ 0

2 Rotatonal quanttes as vectors: RH Rule Pcture apples to: rght-handed coordnates dsplacement angular velocty angular acceleraton α cross product axb torque τrf angular momentum lrp electro-magsm Curl fngers of the rght hand n the sense of the rotatonal moton Thumb shows drecton of a rotatonal vector quantty, perpendcular to the rotaton plane Cross product represents ths computatonally Eample: trad of unt vectors showng rotaton n -y plane kˆ y Rght Hand Rule appled to cross product Multplyng Vectors (Revew) Dot Product (Scalar Product) two vectors a scalar measures the component of one vector along the other a b ab cos( ) a b + a b + a b b a y y dot products of Cartesan unt vectors: ˆ ˆj ˆ kˆ ˆj kˆ 0 ˆ ˆ ˆj ˆj kˆ kˆ a Cross Product (Vector Product) two vectors a thrd vector normal to the plane they defne measures the component of one vector normal to the other smaller angle between the vectors c a b ab sn( ) c a a b b a cross product of any parallel vectors ero cross product s a mamum for perpendcular vectors cross products of Cartesan unt vectors: î î ĵ ĵ kˆ kˆ 0 kˆ î ĵ ĵ î ĵ kˆ î î kˆ î ĵ kˆ kˆ ĵ j k b b b Both products dstnct from multplyng a vector by a scalar a a

3 More About the Cross Product The quantty AB sn() s equal to the area of the parallelogram formed by A and B The drecton of C s perpendcular to the plane formed by A and B The rght-hand rule shows the drecton. The dstrbutve rule: A ( B + C) A B + A C Calculate cross products usng A & B wrtten n terms of the unt vectors (just multply the terms out, or use determnants). A B AyB ABy î + AB AB ĵ + ABy AyB ( ) ( ) ( )kˆ The dervatve of a cross product obeys the chan rule, but preserves the order of the terms: d ( ) d A + d B A B B A Torque as a Cross Product τ r F The torque s the cross product of a force vector wth the poston vector to t s pont of applcaton. τ r F sn( ) r F r F The torque vector s perpendcular to the plane formed by the poston vector and the force vector (e.g., magne drawng them tal-to-tal) Rght Hand Rule: curl fngers from r to F, thumb ponts along torque. Superposton: p τ τ r F all all (vector sum) Can have multple forces appled at multple ponts. Drecton of τ s angular acceleraton as 3

4 Calculatng cross products usng unt vectors Fnd: Soluton: A B Where: A ˆ+ 3 ˆj; B ˆ+ ˆj A B (ˆ+ 3 ˆj) ( ˆ+ ˆj) ˆ ( ˆ ) + ˆ ˆ j + 3 ˆ j ( ˆ ) + 3ˆ j ˆ j 0+ 4kˆ + 3kˆ + 0 7kˆ Calculate torque F (.00ˆ ˆj) N j gven a force and r (4.00ˆ ˆj) m ts locaton: Soluton: τ r F [(4.00ˆ ˆj)N] [(.00ˆ ˆj)m] [(4.00)(.00) ˆ ˆ+ (4.00)(3.00) ˆ ˆj + (5.00)(.00) ˆj ˆ+ (5.00)(3.00) ( )ĵ ˆ ˆjĵ.0kˆ N m k More eamples: cross product usng unt vector epansons Fnd the magntude and drecton of the torque about the orgn for forces appled at pont (0, -4.0 m, 3.0 m) due to: a) force F wth components F.0 N and F y F 0, and (b) force F wth components F 0, F y.0 N, and F 4.0 N? τ r F r 0 î 4 ĵ + 3kˆ F a) î τ τ ˆ ˆ ˆ ˆ ˆ ˆ ˆ r F ( 4 j + 3k ) ( ) ( 4 ) j + ( 3 )k φ kˆ note: ĵ î kˆ, kˆ î + ĵ y τ 6 ĵ + 8kˆ n y- plane î ĵ τ N.m - φ tan ( τ / τ y ) - 0 tan ( 8 / 6) 53 b) F ĵj + 4kˆ k τ r F ( 4 ĵ + 3kˆ ) ( ĵ + 4kˆ ) ( 4 ) ĵ ĵ ( 4 4) ĵ kˆ + ( 3 )kˆ ĵ + ( 3 4)kˆ kˆ note: ĵ ĵ kˆ kˆ 0, ĵ kˆ + î, kˆ ĵ î τ τ N.m î along negatve - as 4

5 Net torque eample: multple forces at a sngle pont 3 forces appled at pont r : r r cos( ) î + 0 ĵ + r sn( ) kˆ F î F kˆ F ĵ r Fnd the torque about the orgn: τ r F r ( F + F + F3 ) (r î + r kˆ) (î + ĵ + kˆ) r î î + r î ĵ + r î kˆ + τ 0 + rkˆ + r ( )ĵ + r ĵ + r ( )î + τ 3 ĵ. j + 5kˆ 5.k o r kˆ î + r kˆ ĵ + set 0 oblque rotaton as through orgn F F r F 3 r rsn( ) 3sn(30 ). 5 o r rcos( ) 3cos(30 ). 6 r kˆ kˆ Here all forces were appled at the same pont. For forces appled at dfferent ponts, frst calculate the ndvdual torques, then add them as vectors,.e., use: τ τ r F (vector sum) all all j o k y Fndng a cross product 5.. A partcle located at the poston vector r (î + ĵ) (n meters) has a force Fˆ ( î + 3ĵ) N actng on t. The torque n N.m about the orgn s? A) kˆ B) 5 kˆ C) - kˆ D) - 5 kˆ E) î + 3ĵ What f Fˆ ( 3 î + 3ĵ)? 5

6 Angular momentum concepts & defnton How much lnear or rotatonal stayng power does a movng object have? - Form the product of an nerta measure and a speed measure. - near momentum: p mv (lnear). What f mass center of object s not movng, but t s rotatng? - Rotatonal (angular momentum): - moment of nerta angular velocty I - lnear momentum moment arm about some as for smple cases lnear rotatonal nerta speed lnear momentum m v pmv I I rgd body angular momentum the angular momentum of a rgd body relatve to a selected as about whch I and are measured: unts: [kg.m /s] I Angular momentum concepts & defnton Eample: angular momentum of a pont mass movng n a straght lne choose pont P as a rotaton as P I mr mvr pr v r r lnear momentum X moment arm v Note: 0 f moment arm 0 s the same for all pont along lne of v Eample: same as above, but wth velocty not perpendcular to r; v rad does not affect P r I mr mv r mvr sn( φ) r p v v v lnear momentum X moment arm Note: 0 f v s parallel to r (radally n or out) φ Eample: angular momentum of a rotatng hoop about symmetry as through P P r v I If t s a hoop: If t s a dsc: I mr v r mr mvr pr same as a pont partcle, for hoop lnear momentum X moment arm I mr mr /r 6

7 Eample: Calculatng Angular Momentum for a Rgd Body Calculate the angular momentum of a 0 kg dsc when: Soluton: 30 rad / s, r 9 cm 0.09 m, m I I mr for a dsk 4 mr kg.96 3 Kg m / s What angular speed would a 0 kg SOID SPHERE have f t s angular momentum s the same as above? Soluton:.96 Kg m /s, r 9 cm 0.09 m, m / I I rad / s 5 mr 0 9 for a sphere kg Angular momentum defned usng cross product The nstantaneous angular momentum of a partcle relatve to the orgn O s defned as the cross product of the partcle s nstantaneous poston vector r and ts nstantaneous lnear momentum p r p m(r v) r p 7

8 Angular momentum of a sngle partcle - defnton etenson of lnear momentum p mv depends on chosen rotaton as (here along ) moment of momentum r p p r pr sn( ) same pcture as for torques use moment arm r r sn().or... tangental momentum component p p sn() only the tangental momentum component contrbutes r and p tal-to-tal always form a plane s perpendcular to that plane moment arm lnear r p m(r v) momentum y P 90 o p r moment arm for p r p 90 o p rad lne of acton of momentum p Conventon: vector up out of paper vector down nto paper (tal) Eample: Angular momentum of a partcle n unform crcular moton The angular momentum vector ponts out of the dagram The magntude s mvr sn (90 o) mvr sn (90 o) s used snce v s perpendcular to r A partcle n unform crcular moton has a constant angular momentum about an as through the center of ts path Eamples: satelltes n crcular orbts O Superposton: angular momenta add as vectors n r p for ths case: + r p - r p all all 8

9 Eample: calculatng angular momentum for partcles PP060-3*: Two objects are movng as shown n the fgure. What s ther total angular momentum about pont O? m m Angular momentum for car 5.. A car of mass 000 kg moves wth a speed of 50 m/s on a crcular track of radus 00 m. What s the magntude of ts angular momentum (n kg m /s) relatve to the center of the race track (pont P )? A) B) C) A D) B E) P 5.3. What would the angular momentum about pont P be f the car leaves the track at A and ends up at pont B wth the same velocty? r p p r pr sn( ) 9

10 Another way to epress the rotatonal Second aw Apply a torque to a wheel: What happens? The torque causes the angular momentum to change Analogous to the relaton of force to lnear momentum,.e., The force actng on a body s the tme rate of change of t s lnear momentum The torque on a partcle equals the tme rate of change of the partcle s angular momentum d τ Ths s the rotatonal analog of Newton s Second aw τ and to be measured about the same orgn The orgn should be n an nertal frame;.e., not acceleratng The Second aw usng angular momentum: sngle partcle dp d near: F Angular analog: τ Proof: start wth the cross-product defnton of angular momentum d d d ( r p) m ( r v) snce p mv epand usng dervatve chan rule: d dr dv m v r m [ v v + r a] + 0 d r ma r F τ Conservaton aws near dp F p s constant f F 0 Momentum Angular d Momentum τ s constant f τ 0 0

11 Translaton Force F near Momentum p mv SUMMARY Rotaton Torque τ r F Angular l r p Momentum Kc Energy K mv Kc Energy K Ι near Momentum Second aw Systems and Rgd Bodes P p Mv Angular cm Momentum F dp Ι for rgd bodes about common fed as Second aw τ d sys Momentum conservaton - for closed, solated systems P sys constant sys constant Apply separately to, y, aes Optonal: Usng determnants for the cross product The cross product can be epressed as ˆ ˆj kˆ Ay A ˆ A A A ˆ Ay A B A ˆ Ay A j+ k By B B B B By B B B y Epandng the determnants gves A B ˆ ˆ j + k ˆ ( A yb A B y ) ( A B A B ) j ( A B y A yb )

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