l. A damped pendulum is described by the (ordinary) dilferential equation [:39 2, (£9
E ——L:.! smB—kE where I? = 9(t) is its angle (with the downward vertical) as a function of t, w = 1 f Q} L
(g = gravitational acceleration, L = length), and k: 13 0 is a damping constant.
Denoting p = #3,, this equation can be re-written as the (ﬁrst-order) system dB at
— = F(E',p), E1” = G(E',p), where F(9,p) =p, C(9,p) = —w2sin9 — kp. (a) i. Verify that 9(t) = 0 and p(t) = 0 (for all t) is a solution, and describe what
it means physically.
ii. Derive an approximate system for 9 close to 0 and p close to 0 by replacing
F(6',p) and G(9,p) with their linear approximations at (9,3)) = (0,0).
iii. What are the solutions of your approximate system if k = 0 (no damping)?
Interpret them physically.
iv. 1What are the solutions of your approximate system if m = 0 (no gravity)?
Interpret them physically.
(b) i. Verify that 6(t) = 11‘ and 19(3) = 0 (for all t) is a solution, and describe what
it means physically.
ii. Derive an approximate system for 6‘ close to er and p close to 0 by replacing
F(r9,p) and G(9,p) with their linear approximations at (9,3?) = (1130). iii. 1What are the solutions of your approximate system if k = 0 (no damping)?
Interpret them physically.