# Find the amplitude and period of the function

1. Apr 29, 2004

### jen043081

A weight hanging on a vertical spring is set in motion with a downward velocity of 6 cm/sec from its equilibrium position. Assume that the constant w for this particular spring and weight combination is 2. Write the formula that gives the location of the weight in centimeters as a function of the time t in seconds. Find the amplitude and period of the function and sketch its graph for t in the interval [0,2(pie)][/i][/b][/tex][/list][/code]

2. Apr 29, 2004

### jepe

Assume the vertical location of the weight from its equilibrium position is z
(z pointing upward is positive)
The mass-spring system can be described as a harmonic motion:
z = za . sin (ω t), where za is the amplitude of the motion
differentiation to t gives the velocity of the mass:
z' = za . ω . cos (ω t)
at t=0: z = 0 cm and z' = -6 cm/s and ω = 2 rad/s
at t=0: -6 = za .2, so za = -3 cm.
since amplitude is always positive, assume za = +3 and write equation of motion of the weight as:
z = -za . sin (ω t) or z = -3 sin (2t)
the period of the harmonic motion is T, where T = 2.pi/ ω
T = pi = 3.1416 s.