Answer: The position of the spring as a function of time is given by: x(t) = 6 * sin(π/4 * t)
where t is the time in seconds and x(t) is the position of the spring in meters.
Step-by-step explanation:
The position of a spring as a function of time can be described by a sinusoidal function. The general form of such a function is:
x(t) = A * sin(ωt + φ) + x₀
where:
x(t) is the position of the spring at time t
A is the amplitude of the motion (maximum displacement)
ω is the angular frequency (related to the period T by ω = 2π/T)
φ is the phase angle (determines the starting point of the motion)
x₀ is the equilibrium position of the spring (where it would be at rest)
In this case, we know that the maximum displacement (amplitude) of the spring is 6m and the period T is 8s. Therefore, we can calculate the angular frequency ω as follows:
ω = 2π/T
ω = 2π/8
ω = π/4
We also know that the spring is at its equilibrium position when t = 0 (i.e., x(0) = x₀). Therefore, we can set x₀ to 0.
Finally, we need to determine the phase angle φ. This can be a bit tricky without more information, as there are many possible starting points for the motion that would produce a sinusoidal function with the given amplitude and period. For simplicity, we will assume that the spring is at its maximum displacement (positive direction) when t = 0. This means that the phase angle φ is 0.
Putting all of this together, we get:
x(t) = 6 * sin(π/4 * t)
This is the position of the spring as a function of time. It describes a sinusoidal motion with an amplitude of 6m and a period of 8s. The motion starts at the maximum displacement (positive direction) and oscillates back and forth around the equilibrium position (0).