Phys 2214 hw2 Solutions - Phys 2214 Homework #2 Solutions...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Phys 2214 Homework #2 Solutions September 9, 2011 1. Free oscillations (a) For the initial conditions x (0) = A and v (0) = 0, we have the following results: i. Underdamped: x ( t ) = Ae- t/ A cos ( t ) ii. Overdamped: x ( t ) = Ae- t/ A iii. Critically damped: x ( t ) = A (1 + t/ A ) e- t/ A These are plotted in the figure. Critically Damped Underdamped Overdamped 2 4 6 8 10- 1.0- 0.5 0.0 0.5 1.0 (b) For the underdamped case, since b 2 4 mk , =- b b 2- 4 mk 2 m 1 becomes =- b i 4 mk- b 2 2 m . The imaginary part of is Im( ) = 4 mk- b 2 2 m = q 2- 1 / 2 A . Since decreases when A 6 = 0, write -| | = q 1- 1 / ( A ) 2 . Dividing by , squaring both sides, and solving for A gives A = 1 q 2 | | /- ( | | / ) 2 . i. For | | / = 0 . 001, A = 22 . 4. ii. For | | / = 0 . 01, A = 7 . 09. iii. | | / = 0 . 1, A = 2 . 29. (c) For the underdamped case, the system undergoes many oscilla- tions during the time it takes to relax to its equilibrium position. For the critically damped and the overdamped cases, the system doesnt undergo any oscillations. i. Auto suspension: critically damped; slightly underdamped. ii. Door bell: underdamped. iii. Speaker cone: underdamped. iv. Pogo stick: underdamped. v. Drum head: critical or underdamped. vi. Cymbal: underdamped. vii. Bungee cord + person: underdamped. 2. Driven oscillations A ( D ) = F /m 2- 2 D + i 2 D / A (a) To find the magnitude, use | A ( D ) | = q A * ( D ) A ( D ) = F m 1 q ( 2- 2 D ) 2 + 4 2 D / 2 A 2 To find the phase, rewrite A ( D ) as A ( D ) = F m 1 2- 2 D + i 2 D / A 2- 2 D- i 2 D / A 2- 2 D- i 2 D / A = F m 2- 2 D- i 2...
View Full Document

Page1 / 8

Phys 2214 hw2 Solutions - Phys 2214 Homework #2 Solutions...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online