Applications of Linear Equations with Constant Coefficients 291
- When a 1 kg mass is attached to a certain spring and the system set
in motion, the period of oscillation is 2 seconds. Later the 1 kg mass is
replaced with an unknown mass m. The new system is set in motion
and the new period is found to be 4 seconds. What is the mass m?
- A .1 kg mass is attached to a spring, the other end is attached to
a fixed support, and the system is allowed to come to rest. In the
equilibrium position the spring is stretched .05 m. Determine if the
spring-mass system will execute damped oscillatory motion, critically
damped motion, or overdamped motion for the following values of the
damping constant
a. c = 2.8 kg/s b. c = 1.6 kg/s c. c = 3.5 kg/s
(HINT: Use POLYRTS or your computer software to find the roots of
the auxiliary equation and based on the type of roots determine the type
of motion.)
- A .8 kg mass is attached to one end of a spring. The other end is
attached to the ceiling. The system is allowed to come to rest. In the
equilibrium position the spring is stretched .3 m. The system is to be set
in motion by pulling the mass downward and releasing it. What value
of the damping constant c will result in damped oscillatory motion?
critically damped motion? overdamped motion?
- A .6 kg mass is attached to the lower end of a spring. The upper end
is attached to a fixed support. When the system is set in motion and
there is no damping the period of oscillation is 2 seconds. When the
damping constant is c kg/s and the system is set in motion the period
of oscillation is 4 seconds. Determine the damping constant c and the
spring constant k.
- For the following RLC series circuits the electromotive force is zero,
E = 0. In each case, determine if the equation for the charge on the
capacitor, q, represents simple harmonic motion, damped oscillatory
motion, critically damped motion, overdamped motion, or none of these.
R (Ohms) L (Henry) C (Coulomb)
a. 0 0.5 2.0 x 10-^5
b. 10 0.0 3.0 x 10-^4
c. 20 0.1 1.0 x 10-^3
d. 20 0.1 0.5 x 10-^3
e. 30 0.2 1.0 x 10-^2
- If E = 0 in an RLC series circuit, what relationship between the
parameters R, L, and C results in an equation for the charge on the
capacitor which represents simple harmonic motion? damped oscilla-
tory motion? critically damped motion? overdamped motion?
