A mass suspended on a spring will exhibit sinusoidal motion when it moves. If the mass on a spring is 85 cm off the ground at its highest position and 41 cm off the ground at its lowest position and takes 3.0 s to go from the top to the bottom and back again, determine an equation to model the data.

Question

A mass suspended on a spring will exhibit sinusoidal motion when it moves. If the mass on a spring is 85 cm off the ground at its highest position and 41 cm off the ground at its lowest position and takes 3.0 s to go from the top to the bottom and back again, determine an equation to model the data.

Answer 1

range = 85-41 = 44

so a = 22
period = 2π/k
k = 2π/3

so basic curve could be
y = 22 sin (2π/3 t) + ?
this would have a minimum of -22, but our min is to be 41, so we have to raise it up 63

y = 22 sin (2π/3*t) + 63

looking ok so far .....
https://www.wolframalpha.com/input/?i=plot+y+%3D+22+sin+%282%CF%80%2F3*t%29+%2B+63

You didn't say if you wanted the lowest point to be when t = 0
If you do, you must find the needed phase shift

Answer 2

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.