Question

A weight is hung from a spring and set in motion so that it moves up and down continuously. The velocity

v
of the weight at any time
t
is given by the equation
v
=
3
cos
(
2
π
t
)

Answer 1

A weight moving up and down on a spring exhibits simple harmonic motion with its velocity represented by a cosine function, similar to the horizontal motion of a mass on a spring.

Step-by-step explanation:

The weight hung from a spring and moving continuously up and down can be described using simple harmonic motion, similar to horizontal spring systems. When considering vertical motion, the force of gravity only shifts the equilibrium position but does not affect the form of the motion. The given velocity equation v=3cos(2πt) indicates that the weight's velocity follows a cosine function, representative of simple harmonic motion. The displacement of the weight can be modeled by y(t) = Acos(wt + p), where A is the amplitude, w is the angular frequency, and p is the phase shift. The phase shift allows the use of either cosine or sine functions for modeling, due to their relationship differing only by a phase shift. At any given time t, the velocity v(t) and acceleration a(t) can be derived from the displacement function, allowing for complete characterization of the simple harmonic motion.