Find the amplitude and period of the function

[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]

 

[jepe]

I assume that w in your thread means the circular frequency ω (in rad/s)?

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.

 

[M98Ranger]

The formula for the location of the weight on the spring as a function of time is given by x(t) = A cos(wt + p), where A is the amplitude, w is the angular frequency, and p is the phase shift. In this case, the weight is set in motion with a downward velocity of 6 cm/sec, which means that the initial displacement is -6 cm. Therefore, the equation becomes x(t) = -6 cos(2t + p).

The amplitude of this function is 6 cm, as it represents the maximum displacement of the weight from its equilibrium position. The period of the function is given by T = 2(pie)/w, where w is the angular frequency. In this case, w = 2, so the period is T = 2(pie)/2 = (pie) seconds.

The graph of this function within the given interval [0,2(pie)] will start at -6 cm (since the initial displacement is -6 cm), reach a maximum value of 6 cm, then return to -6 cm at the end of the period. The graph will be a cosine curve, starting at the origin and oscillating between -6 cm and 6 cm with a period of (pie) seconds. The phase shift, p, will determine where the curve starts within the interval [0,2(pie)].