c Announcement day, ober 8, 004 Ch 8: Ch 10: Work done by orce at an angle Power Rotatonal Knematc angular dplacement angular velocty angular acceleraton Wedneday, 8-9 pm n NSC 118/119 Sunday, 6:30-8 pm n CCLIR 468 Announcement Ch 8: Conervaton o Energy Workheet #1 h week lab wll be a workhop o phyc. Brng your rankng tak book. No quz. CQ1: prng Let now look at a orce that appled at an angle to the drecton o moton. How doe our problem change? I the block reman on the ce, and the man keep the angle between the rope and the ce contant, and the man exert a contant orce, the work done on the block. Skp to reult dtance = Δ Frctonle pond o ce Artocrat at a xed angle Frctonle pond o ce con t 1
We look at the component o the orce that n the drecton o moton o the object (.e., along the drecton o ) -drecton ŝ W = F Δ = Δ = ( co)δ Skp dot product -drecton ŝ = co dtance = Δ dtance = Δ Frctonle pond o ce Frctonle pond o ce W = F Δ Notce: Work a calar quantty, whch mean that you do NO pecy a drecton aocated wth work. Work only ha a magntude. Let ntroduce a new mathematcal operaton to expre the type o product we need to calculate n order to correctly compute work. he calar (or dot) product o any two vector where and A B and A = a x ˆx + a y ŷ + a z ẑ B = b x ˆx + b y ŷ + b z ẑ gven by the expreon A B = a x b x + a y b y + a z b z Whch mathematcally ay: - multply the x-component o the two vector; - add the reult to the product o the y-component o the two vector; and nally - add the reult to the product o the z-component o the two vector. kp I we look at the dtrbutve law o multplcaton n evaluatng our dot product, we dcover everal mportant reult. A B = (a x ˆx + a y ŷ + a z ẑ) (b x ˆx + b y ŷ + b z ẑ) = a x ˆx (b x ˆx + b y ŷ + b z ẑ) + a y ŷ (b x ˆx + b y ŷ + b z ẑ) + a z ẑ (b x ˆx + b y ŷ + b z ẑ) = a x b x + a y b y + a z b z con t
h reult tell u that the ollowng relatonhp mut be true: a x ˆx b x ˆx = a x b x a x ˆx b y ŷ = 0 a x ˆx b z ŷ = 0 Or generally... ˆx ˆx = 1 ˆx ŷ = 0 ˆx ẑ = 0 We can analyze the other two term (or the y- and z- dmenon o A) and nd ŷ ẑ = 0 ŷ ŷ = 1 ŷ ẑ = 0 ẑ ˆx = 0 ẑ ŷ = 0 ẑ ẑ = 1 What alo clear rom our denton o th operaton that calar product are commutatve. A B = B A A B = a x b x + a y b y + a z b z B A = b x a x + b y a y + b z a z Fnally, the calar product alo obey the dtrbutve law o multplcaton. A ( B + C) = A B + A C I leave th a an exerce to you to very on your own. he proo rather mple and traghtorward. dtrbutve Scalar product can alo be nterpreted a the calar product o length o one vector and the length o the projecton o the econd vector onto the rt. B B co A AB co A B = A B co Now, I ve gven you two very derent lookng expreon or the calar product o vector. A B = a x b x + a y b y + a z b z A B = A B co Whch one correct? orm o calar product 3
(he general proo o th a ubject or lnear algebra, but here a pecc cae that llutrate the thnkng.) kp Ung the nd denton o the calar product B B co A For any two -D vector, I can chooe my coordnate ytem uch that one o the vector le entrely along the x-ax. Let chooe vector A to le on the x-ax. ( ) A B = A x B co = A x B x + B y A B = A x B x B x B x + B y Ung the 1t denton o the calar product B A B = A x B x + A y B y B co A B = A x B x A hee two denton o the calar product provde a powerul tool wth whch we can determne the angle between any two vector! Fnd the angle between the ollowng two vector: A = ˆx + 3ŷ Workheet #1a B = 1ˆx + ŷ Fnd the angle between the ollowng two vector: A = ˆx + 3ŷ A B = ()( 1) + (3)() = 4 A B co = + 3 B = 1ˆx + ŷ ( )( ( 1) + ) co A B co = ( 65)co co = 4 / 65 = 60.3 o Now recall that our denton o work W = F Δ W = ( F co ) Δ where the angle between the orce (F) and the dplacement (Δ) 4
W = ( F co ) Δ he rate at whch work done. h expreon, however, jut that o a calar product between the orce vector and the dplacement vector! P = ΔW So, a nce, horthand way to expre work W = F Δ Vald only the orce contant! Work (J) me () Average power Notce the average power output mply the lope o the chord! (Look amlar?) Work (J) Intantaneou power me () We can alo calculate the ntantaneou power beng delvered by a orce. P(t) = lm 0 Average power ΔW = F Δ = F v(t) Notce that Power alo a calar quantty. P = F v [P] = [F][v] [P] = Nm / [P] = J / = W Unt: power A 600 kg elevator tart rom ret and pulled upward by a motor wth a contant acceleraton o m/ or 3 econd. What the average power output o the motor durng th tme perod? 1) 59,90 W ) 1,40 W 3) 17,640 W 4) 3,600 W Workheet # FBD: W Rotatonal Knematc We pent the rt 8 week o cla examnng the moton o pont ma object. We now look more careully at real, extended object that move along curved path and the orce reponble or ther moton.? Elevator motor power (W) 5
o make our le mpler, we re gong to ue the angular unt o meaure known a the radan when dcung the moton o object n crcular arc. Let tart wth the ollowng queton: What π? We all probably remember the numercal value o π (3.14159 ). But rom where doe th numercal value come? Let pend a lttle tme motvatng our choce o radan over degree a the unt o choce or meaurng angle. d π exactly the rato o the crcumerence o a crcle to t dameter. rue or ANY crcle! And we plotted the crcumerence o a varety o crcle agant ther dameter, we would ee Crcumerence Slope = π C = πd Crcumerence Slope = π C = πr Radu Notce our graph now twce a teep a t wa beore. Dameter What we changed to a plot o crcumerence veru radu? How would the plot change? Okay, th tu all SEEMS pretty obvou. Jut where are we headed? How do we compute ARC-LENGH along a crcle? r = r Where meaured n radan! r = Notce that th property namely that the value o quantty /r at a gven doe not depend upon the ze o the crcle property hared wth the trgonometrc uncton (ne, cone, tangent, etc.). How doe the rato o /r change a we ncreae the ze o our crcle? No Change! r co = x 1 r 1 = x x 1 r y 1 r r 1 x 1 y NOE: he angle tll the ame. 6
Here meaured n radan. Notce that the rato /r vare lnearly wth the ze o the angle. r = he meaure o radan provde a natural ytem wth whch to meaure angle. h n contrat wth the ytem o degree, whch reulted rom the completely arbtrary decon to put 360 o them n one complete revoluton around a crcle. Now that we ve een that the natural ytem n whch to meaure n the angular world the radan ytem, we are ree to explore angular moton. Completely analogou to our dcuon o the lnear moton quantte o dplacement velocty acceleraton We can now dene the angular quantte angular dplacement angular velocty angular acceleraton Potve change correpond to counter-clockwe rotaton. r Notce that the angular dplacement can be drectly tranlated nto a lnear dtance traveled ung our arc length rom beore... Δ = = rδ = r( ) r And we know t take the yellow ball a tme to move through the angle Δ, then we can calculate an angular velocty... lope = ω Intantaneou angular velocty Jut a wth t lnear counterpart, we can examne a graphcal repreentaton o th quantty to better undertand t meanng. Δ ω = = t t Angular velocty denoted by the ymbol ω. me lope = ω = Δ Average angular velocty 7
Workheet #3 ω = Δ [ω] = [Δ] [] [ω] = rad = 1 Unt: angular velocty CQ3: Angular velocty Jut a n the lnear cae, where we examned the rate o change o the velocty, n the angular cae, we can examne the rate o change o the angular velocty, whch we call the angular acceleraton. Δω ω ω α = = t t Ang. acceleraton Angular acceleraton denoted by the ymbol α. ω lope = α Intantaneou angular acceleraton me A we dd or the lnear quantty and or angular velocty, let look at the graph. lope = α = Δ ω graphcal nterpretaton Average angular acceleraton α = Δ ω [ α] = [Δ ω] [] [α] = rad/ = Unt: Angular acceleraton Equaton or Sytem Involvng Rotatonal Moton wth Contant Angular Acceleraton Agan, thee are completely analogou to what we derved or the knetc equaton o a lnear ytem wth contant lnear acceleraton! 1 = 0 + ω 0t + αt ω = ω + α 0 t Rot. Knematc equaton 8
A wheel rotate wth contant angular acceleraton o α 0 = rad/. I the wheel tart rom ret, how many revoluton doe t make n 10? Workheet #4 We ve een how arc length relate to an angle wept out: = rδ = r( ) Let look at how our angular velocty and acceleraton relate to lnear quantte. For an object movng n a crcle wth a contant lnear peed (a contant angular velocty), the ntantaneou velocty vector are alway tangent to the crcle o moton. he magntude o the tangental velocty can be ound rom our relatonhp o arc length to angle... = rδ = r Δ v = tan r ω wo wheel, A and B, are rotated wth contant angular acceleraton o α = rad/. Both wheel tart rom ret. I the radu o wheel A twce the radu o wheel B, how doe the angular velocty o wheel A compare to wheel B at tme t = 10? 1) 1/4 a great ) 1/ a great 3) Same 4) tme a great 5) 4 tme a great Workheet #5 wo wheel, A and B, are rotated wth contant angular acceleraton o α = rad/. Both wheel tart rom ret. I the radu o wheel A twce the radu o wheel B, how doe the tangental velocty o wheel A compare to wheel B at tme t = 10? 1) 1/4 a great ) 1/ a great 3) Same 4) tme a great 5) 4 tme a great Workheet #6 For an object movng around a crcle wth a changng angular velocty, and hence a changng tangental velocty, the ntantaneou tangental acceleraton NON-ZERO! Let look at how the tangental velocty change wth tme n uch a cae: vtan = rω Δv tan = v tan v tan t t = rω rω t t 9
For an object movng around a crcle wth a changng angular velocty, and hence a changng tangental velocty, the ntantaneou tangental acceleraton NON-ZERO! Let look at how the tangental velocty change wth tme n uch a cae: r a = Δω tan = r α he tangental acceleraton, however, not the only acceleraton we need to conder n problem o crcular moton... he purple arrow repreent the drecton o the CENRIPEAL ACCELERAION, whch alway pont toward the center o the crcle. Recall our denton o centrpetal acceleraton: a c vt ( rω) = = = rω r r Notce: Our new orm ue angular velocty! 10