Question

A 16 lb weight stretches a spring 6 inches in equilibrium. It is attached to a damping mechanism with constant c. Find all values of c such that the free vibration of the weight has infinitely many oscillations.

Answer 1

Answer:

$-8<c<8$

Explanation:

Given data:

Weight W = 16 lb

length l = 6/12 = 0.5 ft

hence, spring constant  k = W/l = 16/0.5 = 32 lb/ft

The equation of motion of spring is

$mx''+cx+kx=0\\0.5x''+cx'+32x=0$

the auxiliary equation can be written as

$0.5r^2+cr+32=0\\r^2+2cr+64=0$

The discriminate of equation is

$D =(2c)^2+-4(1)(64)\\=4(c^2-64)$

To get the value of the damping constant,

$D<0\\4(c^2-64)<0\\c>-8, c<8\\-8<c<8$