5 An object stretches a spring 5 cm in equilibrium It is ini

5. An object stretches a spring 5 cm in equilibrium. It is initially displaced 10 cm above equilibrium and given an upward velocity of .25 m/s. Find and graph its displacement for t > 0. Find the frequency, period, amplitude, and phase angle of the motion.

Solution

Suppose mass of object is m then

mg =k (0.05)

196m = k   ........(i)

m w^2 = k

m w^2 = 196m

w = 14 rad/s

w = 2pi f = 14

f = 2.228 Hz ..............Ans (frequency )

T = 1/f = 0.45 Sec .....Ans

for amplitude,

total energy = PE + KE at any moment = k A^2 /2

at t= 0

total energy = k (0.10^2) /2 + m (0.25^2) /2

      =( 0.01k + 0.0625m) / 2   = k A^2 /2

0.01k + (0.0625)(k/196) = kA^2

A = 0.102 m = 10.2 cm ..........Ans

Phase angle ( phi) :

x = A sin(wt + phi)

10.2 = 10 cos( 0 + phi)

phi =80 deg ........Ans