To determine an equation that models the given data, we can make use of the properties of a sinusoidal function.
The general equation for a sinusoidal motion can be written as:
y = A * sin(B * (x - C)) + D
Where:
A is the amplitude of the motion,
B is the frequency,
C is the phase shift, and
D is the vertical shift.
Let's break down the given information and find the values for these parameters:
1. Amplitude (A):
The amplitude of the motion can be determined by finding the difference between the highest and lowest positions of the mass.
A = (highest position - lowest position) / 2 = (85 cm - 41 cm) / 2 = 44 cm / 2 = 22 cm.
2. Frequency (B):
The frequency of the motion can be determined by calculating the number of complete cycles (up and down) the mass completes in a given time period. In this case, it takes 3.0 seconds for one complete cycle.
B = 2π / period = 2π / 3.0s ≈ 2.094 radians/s.
3. Phase Shift (C):
The phase shift determines the horizontal position of the sinusoidal graph. Since there is no mention of any phase shift in the question, we assume C = 0.
4. Vertical Shift (D):
The vertical shift determines the mean position of the sinusoidal graph. In this case, the mean position is the mid-point between the highest and lowest positions.
D = (highest position + lowest position) / 2 = (85 cm + 41 cm) / 2 = 126 cm / 2 = 63 cm.
Now that we have determined the values for A, B, C, and D, we can substitute them into the general equation to get the equation that models the data:
y = 22 * sin(2.094 * x) + 63.
So, the equation that models the given data is y = 22 * sin(2.094 * x) + 63.