Answer:
The graph is attached below :
Step-by-step explanation :
For better understanding of the solution, see the graph which is attached below :
The first point will be at the max height not on the mid line.
Now, using the concept of sine tool, the function can be found to be :
y = A·sin(B·x - C) + D
where, Amplitude = A = 10
Period = 12 seconds
[tex]B=\frac{2\cdot \pi}{period}\\\\\implies B=\frac{2\cdot \pi}{12}\\\\\implies B=\frac{\pi}{6}[/tex]
Since at x = 0, the weight is at its highest point
Therefore, At x = 0 the sine graph is shifted back by [tex]\frac{\pi}{2}[/tex] units. Thus,
[tex]C= -\frac{\pi}{2}[/tex]
The mid line of the graph is y = 0
⇒ D = 0
Hence, The function is :
[tex]\bf y=10\cdot \sin((\frac{\pi}{6})\cdot x+\frac{\pi}{2})[/tex]
The graph of the motion of the weight created with a sine function
showing the first point on the midline and second at the next lowest point.
Steps to plot the graph:
Location of the weight at the beginning of the observation = The highest point
The time it takes the weight to complete a cycle, T = 12 seconds
The difference between the lowest and highest point = 10 in.
The assumed location of the resting position is y = 0
The general form of the sine function is y = A·sin(B·x + C) + D
The period, T = [tex]\frac{2 \cdot \pi}{B}[/tex] = 12
Therefore;
[tex]B = \dfrac{2 \cdot \pi}{12} = \mathbf{\dfrac{\pi}{6}}[/tex]
A = The amplitude
D = The vertical shift = 0
The difference between the highest and lowest point = 2 × A
[tex]A = \dfrac{10}{2} = 5[/tex]
At x = 0, the function is at the highest point, therefore;
sin(B×0 + C) = 1
sin(C) = 1
Therefore
[tex]C = \dfrac{\pi}{2}[/tex]
The function is therefore;
[tex]y = \mathbf{5 \cdot sin \left(\dfrac{\pi}{6} \cdot x + \dfrac{\pi}{2} \right)}[/tex]
Please find attached the graph of the function plotted with MS Excel,
showing the first point on the midline with coordinates approximately (3,
0), and the second point at the point (6, -5).
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