Question

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(1 point) Suppose a pendulum with length L (meters) has angle 0 (radians) from the vertical. It can be shown that 0 as a function of time satisfies the differential equation: d20 +sin0 0 dt2 where g 9.8 m/sec/sec is the acceleration due to gravity. For small values of 0 we can use the approximation sin(0)~0, and with that substitution, the differential equation becomes linear. A. Determine the equation of motion of a pendulum with length 1 meters and initial angle 0.1 radlans and initial angular velocity de/dt 0.5 radians/sec B. At what time does the pendulum first reach its maximum angle from vertical? (You may want to use an inverse trig function in your answer) seconds c. What is the maximum angle (in radians) from vertical? D. How long after reaching its maximum angle until the pendulum reaches maximum deflection in the other direction? (Hint: where is the next critical point?) seconds E. What is the period of the pendulum, that is the time for one swing back and forth? seconds

Answer 1

Sina in Amall d'o SinG 20 O=O L -9.9/42 A L-1m Glo)= o.1redG'0) 0.5rd/sc 9.8. 0 O + clt2 oro)-0.5 glo)- 0.1 "+9.90- o, Olo-0.1, ro) =0.5 L Compare to "+ wo- o, w 9.8 Gunel Salut af uill hau 0 ACor wt B Sin wt/ TA=. 0.1 Cas wt+B Sinwt →e'--유 Sih (wt) +B w Cos w - 0 Bw 0.5-6(o)

S 2-o.s 2.8 .()0.1 Car) + Sin9-8 D Let A AB Sinp Car p A B2 .8 6.1 A Tan Ta -8 0.5593 824/ S

2 2 87 24 SO . e=7 Sn Cas t+ Cos tSin -) 2450 Sin/ 9.3+ whare/ 24S ETan O.55938 924 il be ma Whn Sin (.8 ) -4 9.8 &7Yad 0./8441S14 -0.140747944 gec ad max To do this, pendala, uill take time 94 ual to half f piod af O 2450 T C TT 2TT ec .8 peied