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for a.ii), I got that

F(theta,p) (approx) = p, and the approximation for G(theta,p) = -w^2*theta-kp (by setting the linear approximations for F and G equal to the actual equations for F and G). From this, I conclude that theta=sin(theta). However, in part iii), I'm not sure how to get solutions of this approximate system (theta=sin(theta)) by setting k=0. Perhaps the approximate system I have is incorrect.

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1. A damped pendulum is described by the (ordinary) differential equation
dgfl 2 (£3 E = —w sinfl — FEE
where H = fit) is its angle [with the domward vertical) as a function of t, w = 1/ g} L
{g = gravitational acceleration, L = length), and k 23 I] is a damping constant. Denoting p = %, this equation can be re-written as the [first-order) system — = PM), — = Gimp). where PM) = :2. 603.12) = —wflsina — rap. (a) i. Verify that fit) = CI and p[t) = U [for all t) is a solution, and describe what
it means physically.
ii. Derive an approximate system for 9 close to CI and 10 close to CI by replacing
PH), 15)) and am, 19) with their linear approximations at (3,153) = [I], 0).
iii. What are the solutions of your approximate system if k = II} [no damping)?
Interpret them physically. iv. 1What are the solutions of your approximate system if m = II} {no gravity)?
Interpret them physically.

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