A researcher observes and records the height of a weight moving up and down on the end of a spring. At the beginning of the observation the weight was at its highest point. From its resting position, it takes 12 seconds for the weight to reach its highest position, fall to its lowest position, and return to its resting position. The difference between the lowest and the highest points is 10 in. Assume the resting position is at y = 0.
The first point must be on the mid-line and the second point must be a maximum or minimum value on the graph closest to the first point.
this is just another sine wave problem. What gives you trouble? How far do you get? Just use each sentence to determine one of the characteristics of the function.
To find the equation that represents the height of the weight as a function of time, we can use the formula for simple harmonic motion when the weight is attached to a spring:
y(t) = A * cos(ωt + φ) + c,
where:
- y(t) is the height of the weight as a function of time,
- A is the amplitude of the motion (half the distance between the highest and lowest points),
- ω is the angular frequency of the motion (related to the period T),
- φ is the phase constant, and
- c is the vertical displacement (the resting position).
Given that the resting position is at y = 0 and the amplitude (A) is 10 inches (half the distance between highest and lowest point), we can rewrite the equation as:
y(t) = 10 * cos(ωt + φ).
We know that from the resting position, it takes 12 seconds for the weight to reach its highest point, fall to its lowest point, and return to its resting position. This means that the time period (T) is 12 seconds.
The relationship between angular frequency (ω) and time period (T) is given by the equation:
ω = 2π / T.
Substituting the value of T into the equation:
ω = 2π / 12 = π / 6.
Now we have the angular frequency (ω) value.
To determine the phase constant (φ), we are given that the first point must be on the mid-line (y = 0) and the second point must be a maximum or minimum value on the graph closest to the first point. Since the weight starts from its highest point and cycles back to the resting position, the first point must correspond to the maximum value.
At the maximum point, y(t) = A = 10 inches. Substituting this value into the equation:
10 = 10 * cos(ωt + φ).
Simplifying the equation:
cos(ωt + φ) = 1.
Since the maximum value of cosine is 1, it implies that ωt + φ = 0 when y(t) is at its highest point. Therefore:
ωt + φ = 0.
Substituting the value of ω:
(π/6)t + φ = 0.
Solving for φ:
φ = -π/6.
Now we have the phase constant (φ) value.
Combining all the information, the equation that represents the height of the weight as a function of time is:
y(t) = 10 * cos(π/6 * t - π/6).