Chapter 10. Rotational Kinematics
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1 Chape 10. Roaonal Knemacs Up o now we hae only consdeed ponpacles.e. we hae no consdeed he shape o sze only he mass Also we hae only consdeed he moon o pon-pacles sagh-lne ee-all pojecle moon. Bu eal objecs can also umble wl Ths subjec oaon s wha we exploe n hs chape and n Chape 11. Fs we begn by exendng he conceps o ccula moon
2 Insead o a pon-pacle consde a hn dsk o adus spnnng on s axs y θ z Axs o oaon θ s ac lengh s x θ Ths dsk s a eal objec has sucue We call hese knds o objecs Rgd Bodes Rgd Bodes do no bend ws o lex; o example a bllad ball ac lengh adus s Uns o adans (ad)
3 s θ Fo one complee eoluon s π ccumeence Coneson elaon: π ad 360 θ π ad Now consde he oaon o he dsk om some nal angle θ o a nal angle θ dung some me peod o Δθ θ θ θ θ Δθ Δ ω ag Aeage angula elocy (uns o ad/s) Angula dsplacemen (uns o ad ccw s +) z y θ ccw θ x
4 Smla o nsananeous elocy we can dene he Insananeous Angula Velocy A change n he Angula Velocy ges ω ω ω lm Δ 0 Δθ Δ Δω Δ α ag Aeage Angula Acceleaon (ads/s ) Analogous o Insananeous Angula Velocy he Insananeous Angula Acceleaon s α lm Δ 0 Δω Δ
5 Acually he Angula Veloces and Angula Acceleaon ae magnudes o eco quanes Wha s he decon? ω and α They pon along he axs o oaon wh he sgn deemned by he gh-hand ule Example A an akes.00 s o each s opeang angula speed o 10.0 e/s. Wha s he aeage angula acceleaon (ad/s )?
6 Soluon: Gen:.00 s ω 10.0 e/s Recognze: 0 ω 0 and ha ω needs o be coneed o ad/s ω 10.0 e s 6.8 ad s π ads 1 e α ag ω ω 0.0π α ag 31.4 ad s 0.0π ad s 10.0π ad s
7 Equaons o Roaonal Knemacs Jus as we hae deed a se o equaons o descbe ``lnea o ``anslaonal knemacs we can also oban an analogous se o equaons o oaonal moon Consde coelaon o aables Tanslaonal x dsplacemen θ elocy ω a acceleaon α me Roaonal
8 Replacng each o he anslaonal aables n he anslaonal knemac equaons by he oaonal aables ges he se o oaonal knemac equaons (o consan α) θ θ + 1 (ω + ω )( - ) θ θ + ω ( - ) + 1 α( - ) ω ω + α( - ) ω ω + α(θ -θ ) We can use hese equaons n he same ashon we appled he anslaonal knemac equaons
9 Example Poblem A gue skae s spnnng wh an angula elocy o +15 ad/s. She hen comes o a sop oe a be peod o me. Dung hs me he angula dsplacemen s +5.1 ad. Deemne (a) he aeage angula acceleaon and (b) he me dung whch she comes o es. Soluon: Gen: θ +5.1 ad ω +15 ad/s Ine: θ 0 ω 0 0 Fnd: α?
10 (a) Use las knemac equaon ω ω + α(θ θ ) 0 ω + αθ α ω (15 ad/s) θ (5.1 ad) (b) Use s knemac equaon ad s θ θ θ 0 θ ω ( ω ( ω + (5.1ad) 15 + ω 0)( )( ad/s 0) ) 0.68 s
11 O use he hd knemac equaon ω ω + α( ) 0 ω + α ω α Example Poblem 15 ad/s - ad/s 0.68 s A he local swmmng hole a aoe ck s o un hozonally o a cl ha s 8.3 m aboe he wae uck no a ``ball and oae on he way down o he wae. The aeage angula speed o oaon s 1.6 e/s. Ignong a essance
12 deemne he numbe o eoluons whle on he way down. y y ω Soluon: y x Gen: ω ω 1.6 e/s y 8.3 m Also y 0 0 y 0 Recognze: wo knds o moon; D pojecle moon and oaonal moon wh consan angula elocy. Mehod: #eoluons θ ω. Theeoe need o nd he me o he pojecle moon.
13 Consde y-componen o pojecle moon snce we hae no nomaon abou he x- componen..1 e e/s)(1.3 s) s 9.80 m) s m (1.6 ) )( ( 1.3 ( y y g y g y g y ) - ( a ) - ( y y θ ω ω ω θ θ
14 Tangenal Velocy z ω θ Fo one complee eoluon he angula dsplacemen s π ad Fom Unom Ccula Moon he me o a complee eoluon s a peod T Theeoe he angula elocy (equency) can be wen Δθ π ω ω Δ T (ad/s)
15 Also he speed o an objec n a ccula pah s π ω T Tangenal speed T (m/s) The angenal speed coesponds o he speed o a pon on a gd body a dsance om s cene oang a an angula speed ω Each pon on he gd body oaes a he same angula speed bu s angenal speed depends on s locaon T 0
16 I he angula elocy changes (ω s no consan) hen we hae an angula acceleaon α Fo some pon on a dsk o example Fom he denon o anslaonal acceleaon ω ω α ω ω ω ω a a Δ Δ Δ Tangenal acceleaon (uns o m/s ) Snce he speed changes hs s no Unom Ccula Moon. Also he Tangenal Acceleaon s deen om he Cenpeal Acceleaon.
17 Recall a c We can nd a oal esulan acceleaon a snce a and a ae pependcula a a + a φ an 1 (a /a ) Peously o he case o unom ccula moon a 0 and aa c a. The acceleaon eco poned o he cene o he ccle. I a 0 acceleaon pons away om he cene a z a φ a a a a
18 Example A hn gd od s oang wh a consan angula acceleaon abou an axs ha passes pependculaly hough one o s ends. A one nsan he oal acceleaon eco (adal plus angenal) a he ohe end o he od makes a 60.0 angle wh espec o he od and has a magnude o 15.0 m/s. The od has an angula speed o.00 ad/s a hs nsan. Wha s he od s lengh? Gen: a 15.0 m/s ω.00 ad/s (a some me)
19 z y L a L a α Lα ω Lω a (Lω) a acosφ Lω L acosφ ω x L a φ a Sole o L (15.0 m )cos60.0 s (.00 ad s ) a Lω 1.88 m
20 Rollng Moon (Tes Bllads) Demo 8.6. B A C Road axs ca Consde a e aelng on a oad wh con (no skddng) beween he e and oad Fs eew concep o elae elocy. Wha s he elocy o B as seen by he gound? I e s suspended eey pon on edge has same
21 Pon B has a elocy wh espec o he A Now A (he e as a whole) s mong o he gh wh elocy ca ca ca G B ca A G A B G B ca A G A B + + Wha s elocy o C as seen by he gound? ca G C ca A G A C G C ca A G A C
22 Example Poblem A e has a adus o m and s cene moes owad wh a lnea speed o 15.0 m/s. (a) Wha s ω o he wheel? (b) Relae o he axle wha s o a pon locaed m om he axle? 15.0 ω ca 45.5 ad/s ω (0.175)(45.5) 7.96 m/s
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