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.