hw21 - 13-1954 a particle is fellcwing a path given by r{t}...

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Unformatted text preview: 13-1954 a particle is fellcwing a path given by r{t} = so an; 33 where aft] in in radians and r is in millimeters. Given that 3 = 2.5 radfe {ccnetant) and that 9 = 0 when t = 0. determine the velccity and acceleraticn cf the particle when t = 2 e. Salutiun Given the angular velccity 3 = 2.5 radfe = cenetant, the angular pceiticn and angular acceleraticn are E Differentiating the radial pceiticn r = 50 :55 33 = 50 gas 7_5t mm with respect tc time given r = -375 cin 7.5t mmjc ii 2 r = —2312.5 cue ?.5t mm/e Then when t = 2 s. 9 = 5 red = 234.43n r = —3?.934 mm E- = —243.353 mm/e E = 1375.93 mun/42 é = 2.5 rad/e E = 0 rad/e2 vr = E- = ~243.a5a mmfc v5 = = -94.950 mm/e f i? = —243.353 Er — 94.96:: fie mm/e = 262 mfe a 52.24“ ar = E - réz = 1514.33 w: .43 = :5 + zéé = 41219.25 mum/53 'a' = 1614.33 Er - 1219.25 EB mmfez - I 1. _-1. I .. “If. 141119.?I ."f‘; 1'“. ---'-------------u-u---44~..h."fi-'~Wm _ 5!. 1" ' I l- . :-._ :34 H'.‘ .51 . ’. :_ ': ... . 'I ‘3. “A _ _.f__. 1. '. II- -'- 2 a 2023 mmfe ? 59.42 v cf the cellar. r 9 c. Determine the acceleraticn ccmpcnenta a and a9 cf the ccllar. I' d. Verify that the velccitydrectcr‘v and 1fififlfl the acceleraticn vectcr a are bcth ' directed alcng the hcriacntal rcd. Bcluticn Given the angular pcaiticn cf the arm fl/Z - fl/E cca 1.5t rad, the angular velcci cf the arm are i- 5 = 5111 1.5: rad/e 6 = Ens The radial Pnaitinn “f the Pin is given by .” r sin 3 = 2 ft ‘; _ i; r- 73' 1;. “I. r Bi“ 6 + r3 ‘2“ 9 = D " '. “ ' .. e». r Bin 3 + Zrfl gas a + r6 fine 6 h :32 Einfia = a i g ,Qwflq ' - 'I""I"I5-;i.i1.r. 1_—.._r: . - ' ' _ *3.- .ym' ill”. i. Then at t = 5 a _ _ H_hh;m fl ' - 6 = 79.50 B = D_?367n rafiffl ENGINEERING HECHRHICS - Dynamics 13—112 firm AB cf the cam fcllcwer mechanism shcwn is rctating at a ccnstant angular speed cf w 60 rev/min. A spring helds the pin P against the cam inhes. The eguaticn that describes the shape cf the cam lches is r = 2D + 15 ccs B D where r is in millimeters. When 3 = T5 , a. Determine the velccity ccmpcnents er and VB cf the pin P. b. Determine the acceleraticn ccmpcnents er and a6 cf the pin P. c. Verify that the velccity vectcr fiiis directed alcng the cam surface. I .. fl . .., ‘45-" flail-w - I‘ll-Hindi"- rslfl+15mmfl Ecluticn Since the arm rctates at a ccnstant angular rate {6 = 60 revfmin = 2n radfs}, the angular acceleraticn is serc. Differentiating the expressicn fer the radial pcsiticn cf the pin r = 20 + 15 ccs 3 mm r = —159 sin 3 mmfs n '2 a r=-153 13:156me Then at B = 75°r r = 23.33229 mm E = -91.03635 mmfs " Z I = '153.2565 mme scientist's-nag...__.__-__"._ a 36': 150.1 mmfs .______________'__.__".-li'i' final:ifif "3 ii = 115.5 m/e ? 16.24 ENGINEERING MECHANICS - Dynamics W.F. Riley 5 L.D. Sturges {Problem 13~112 - cont.) The tangent to the surface is found from the slope tan ¢ = dy/dx where r = 20 + 15 cos 5 (= 23.38229 mm} y = r sin 9 {= 23.06852 mm] x = r cos 9 {= 6.18119 mm} dr/de = -15 sin 9 {= -14.48889 mm} dy/dfi = dr/dfi sin 6 + r cos 9 (= —7.31400 mm} dx/dfi : dr/dfi cos 9 — r sin 9 [= -26.31852 mm} : dz = dgfldfi = —?.31400 = tan ¢ dx dx/de -26.81852 0'29137 ¢ = 16.24D which is the same direction as the velocity vector. ...
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This note was uploaded on 04/27/2009 for the course MEEN 221 taught by Professor Mcvay during the Spring '08 term at Texas A&M.

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hw21 - 13-1954 a particle is fellcwing a path given by r{t}...

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