In this problem we will use mathematica to nd the

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Unformatted text preview: all angle assumption the motion is periodic with period τ0 = 2π l/g . In this problem we will use Mathematica to find the period for large oscillations as well: ˙ (a) Using conservation of energy, determine φ as a function of φ, given that the pendulum starts at rest at a starting angle of Φ0 . Now use this ODE to find an expression for the time the pendulum takes to travel from φ = 0 to its maximum value Φ0 . (You will have a formal integral to do that you cannot solve analytically! That’s ok, just write down the integral, being very explicit about the limits of integration) Because this time is a quarter of the period, write an expression for the full period, as a multiple of τ0 . (b) Use MMA to evaluate the integral and plot τ /τ0 for 0 ≤ Φ0 ≤ 3 rad. For small Φ0 , does your graph look like that you expect? (Discuss - what value DO you expect?) What is τ /τ0 for Φ0 = π /8 rad? How about π /2 rad? What happens to τ as the amplitude of the oscillation approaches π ? Explain. Note: The integral has some curious numerical...
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