1550078481-Ordinary_Differential_Equations__Roberts_

290 Ordinary Differential Equations

c. What is the phase angle?

7. A damped p endulum of length .6 m with a bob of mass .5 kg is

started with init ial position y(O) = 0 and initial velocity y' (0)

.2 radians/second in a medium with damping constant c = .5 kg/s.

a. Write the equation of motion. (HINT: Use POLYRTS or your

computer software to find t he roots of the auxiliary equat ion.)

b. What is the maximum angular displacement from the vertical?

8. A pendulum of length .2 m with a bob of mass .5 kg oscillates in a
   viscous medium. Determine if t he pendulum will execute damped oscil-
   latory motion, critically damped mot ion , or overdamped motion for t he
   following damping constants

a. c = 10 kg/s b. c = 7 kg/s c. c = 5 kg/s

(HINT: Use POLYRTS or your computer software to find the roots of

the a uxiliary equation and based on t he type of roots determine the type

of motion.)

9. In an experiment a pendulum of length. 7 m with a bob of mass
   .3 kg oscillating in a viscous medium was observed to execute damped
   oscillatory motion. Two successive maxima angular displacements were
   measured to be y(t 1 ) = 1 /5 radian and y(t 2 ) = 1 /6 radian. Find the
   damping constant, c, of the medium. (HINT: The time interval between

successive maxima, t2 - ti, is 2n/w. Write equation (20) for t 1 and t2,

divide y(t 1 ) by y(t2), and solve for c.)

Exercises 10-13 pertain to spring-mass systems. Assume the mass

attached to the spring is executing simple harmonic motion- that

is, assume there is no damping so c = 0.

10. a. What is the velocity of the mass at the instant the displacement from
    the equilibrium position is a maximum?
    b. What is the position of the mass when the velocity is a maximum?

11. A mass mis attached to a spring whose spring constant is 3.2 kg/s^2 , if

the period of oscillation is 2 seconds, determine the mass m.

12. The top of a spring is attached to a fixed support. A 2 kg mass is

attached to the bottom of the spring. After coming to rest at the equi-

librium position, the mass is pulled down X meters below t he equilibrium

position and released with an initial downward velocity of .3 m/s. If the

amplitude of the resulting harmonic motion is .1 m and the period is

1 s, calculate the spring constant k and t he distance X.