Assignment For Section 5.4 and 5.6

You should read the section before attempting any reading questions or problems. Remember that you are responsible for all definitions and examples.

1) Define the following:

a) dashpot

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): F_{E} (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**