Equation of a simple harmonic motion (Finding the Amplitude and Phase Shift)

Question

Question : A weight is suspended from a spring and is vibrating vertically. Suppose the weight passes through its control position as it is rising when t = 2.5, then attains a maximum upward displacement of 4 cm and passes through its central position as it is descending when t = 3.5. The motion is simple harmonic and described by equation of the form f(t) = a sin (t - c). Where f(t) centimeters is the directed distance of the weight from its central position after t seconds and the positive direction is upward. Find this equation.

Graph from the given values in the problems

My Analysis :

• The amplitude (a) = 4, so the only missing value now is the "c" which tells us about the phase shift.

• The period formula is w = 2*pi / |b|; but the value of "c" nor the "b" is given hence I am stuck.

• The formula for phase shift is C / |b| but both are missing.

From the book :

The solution is w/2 = (7/5) - (5/2). So from that, w (the period) = 2.

The value of w was then substituted to the formula of the period to get "b".
w = 2*pi / |b|; b = 2*pi / 2 = pi.

The final answer is f(t) = a sin b(t-c) = 4 sin pi (t - 5/2) where b = pi and c = 5/2.

My questions :

1. Where did the w/2 = 7/5 - 5/2 came from? What formula is that? The 7/5 and 5/2 is from t = 3.5 and 2.5 s but why is subtracted? Why is it equal to w/2? Where did the w/2 or period over 2 came from?

2. The calculated period is 2 while the "b" is pi. So when the values are substituted to the equation a sin b (t-c), why is c = 5/2?

Any help will be appreciated. Thank you and God bless.

Answer 1

1. Where did the $\frac{w}{2} = \frac{7}{5} - \frac{5}{2}$ came from? What formula is that? The $\frac{7}{5}$ and $\frac{5}{2}$ is from $t = 3.5$ and $2.5$ s but why is subtracted? Why is it equal to $\frac{w}{2}$? Where did the $\frac{w}{2} or period over $2$ came from?

As I'm sure you know, the period is how long it takes for one cycle to occur, in a normal sinusoidal, $\sin(x)$, the period is just $2\pi$. In the problem (and the graph makes this more apparent), we're shown how far it takes to get through half of the period. That's why it's $\frac{w}{2}$. The two values for t are subtracted because we want to find the distance between the two points, A.K.A. how long half of the period is. (just like how the distance between $t=3$ and $t=5$ would be $2$)

2. The calculated period is $2$ while the "b" is pi. So when the values are substituted to the equation a sin b (t-c), why is $c = \frac{5}{2}$?

A great way to think of the phase shift for this is to ask yourself where it'll start. Again, this is much more obvious in the graph, but making the $c=\frac{5}{2}$ is just like saying "This is a sine function, but it starts at $\frac{5}{2}$ instead of $0$".

Hopefully that helps.