Question

True or False Question. Please provide a full explanation.

(m) Suppose we have mass of weight 1kg hanging off a spring with spring constant k = 10 and a damping constant B = 2. This mass is released from rest above the equilibrium location. A free damped motion follows a differential equation y"2y10y 0 This system has no longterm oscillating behaviour (i.e. it is overdamped or critically damped)

Answer 1

considering the differential equation is

m u'' + v u ' + k u = F( t )

for free vibration F ( t ) = 0

and it is un damped equation so r = 0

then

m u'' + v u ' = 0

m r² + k = 0

r = $\pm$ i ( k/m )^1/2

m u '' + $\gamma$ r + k = 0

m r² + $\gamma$ r + k = 0

r = - $\gamma$$\pm$ ( $\gamma$² - 4 m k )^1/2 / 2 m

$\gamma$² - 4 m k = 0

if

$\gamma$² - 4 m k > 0

y '' + 9 y = 0

r² + 9 = 0

r = $\pm$ 3 i

so the behavior is is TRUE

now the given equation

y '' + 2 y ' + 10 y = 0

r² + 2 r + 10 = 0

r = ( - 2 $\pm$ 6 i ) / 2

$\gamma$² - 4 m k

= 2² - 4 x 1 x 10

= - 36 < 0

iti says TRUE.