Assignment For Section 5.4 and 5.6
1) Define the following:
b) damped and undamped motion
c) free and forced Motion
d) critical damping
e) natural and external frequencies
f) transient solution and steady periodic solution
2) On page 312, why do FS and x have opposite signs in equation (1)?
3) On page 312, how do we get Eq(3): FE (t)= mx"+cx'+kx
4) Just for your information, on page 313 the author used the fact that arc length equals the radius (L) times the angle (theta) in radians. A little later the author claims that T+V=constant; why is this true?
5) Prove that sin(t) is approximately equal to t for small angle measurements by showing that the limit of sin(t)/t is 1 as t goes to zero. (Does L'Hopital's ring a bell?)
6) Prove that Eq(10) is the general solution of Eq(8').
7) On page 316-17, recall that c represent the damping constant of the dashpot. Do solutions (18) and (19) exhibit the dynamics you would expect from shock absorbers in the real world? Explain.
8) What are the similarities and differences of 5.4 and 5.6?
9) Fill in the blanks needed to derive Eq(7) on page 340 and then use this to get Eq(9) under the initial conditions given on page 341.
10) Why is the amplitude of xp large when natural and external frequencies are close in value.
11) Derive Eq(11) on page 342.
12) Derive Eq(14) on page 343. Give a real world example of a situation in which the natural frequency is equal to the external frequency.
Problems: page 320 1, 4, 5, 12, 13, 15, 22,
24, 25, 26, 27
page 348 2, 5, 10, 15, 19, 20, 22, 26, 28, 29, 30