# Simple Harmonic Motion: why sin(wt) instead of sin(t)?

• thebosonbreaker

#### thebosonbreaker

Hello,
I have recently been introduced to the topic of simple harmonic motion for the first time (I'm currently an A-level physics student). I feel that I have understood the fundamental ideas behind SHM very well. However, I have one question which has been bugging me and I can't seem to find a valid answer.
I will consider the example of a simple pendulum which has been set in motion and therefore oscillates about a fixed equilibrium position.
If the displacement of the pendulum bob is considered as a function of time, then the graph of x/t is analogous to a sine curve (assuming that the bob is released from a point of maximum displacement - since I believe that starting from the equilibrium position would produce a cosine curve [please correct me if I'm wrong here])
The graph will have the equation x = Asin(wt). Now I try to break this down in order to understand why this equation is true for SHM.
Firstly, as I said the variation of x with t produces a sine curve, explaining why X is a function of sin(t). I'm fine with that. Next, I understand that A (the amplitude or max. displacement) is a coefficient on the outside because it has the effect of 'stretching' the sine curve (since the bob oscillates between positions of max. displacement either side.) Again, that all makes sense. However, what I do not understand is why sin(wt) [I know omega is the letter used but I only have w on my keyboard] is used instead of sin(t). It is time which is plotted on the horizontal axis, so surely the y-axis represents displacement (x) and the x-axis represents time (t).
Why is the equation not:
x = Asin(t)?

If somebody could clear this up for me it would be greatly appreciated.

thebosonbreaker said:
Hello,
I have recently been introduced to the topic of simple harmonic motion for the first time (I'm currently an A-level physics student). I feel that I have understood the fundamental ideas behind SHM very well. However, I have one question which has been bugging me and I can't seem to find a valid answer.
I will consider the example of a simple pendulum which has been set in motion and therefore oscillates about a fixed equilibrium position.
If the displacement of the pendulum bob is considered as a function of time, then the graph of x/t is analogous to a sine curve (assuming that the bob is released from a point of maximum displacement - since I believe that starting from the equilibrium position would produce a cosine curve [please correct me if I'm wrong here])
The graph will have the equation x = Asin(wt). Now I try to break this down in order to understand why this equation is true for SHM.
Firstly, as I said the variation of x with t produces a sine curve, explaining why X is a function of sin(t). I'm fine with that. Next, I understand that A (the amplitude or max. displacement) is a coefficient on the outside because it has the effect of 'stretching' the sine curve (since the bob oscillates between positions of max. displacement either side.) Again, that all makes sense. However, what I do not understand is why sin(wt) [I know omega is the letter used but I only have w on my keyboard] is used instead of sin(t). It is time which is plotted on the horizontal axis, so surely the y-axis represents displacement (x) and the x-axis represents time (t).
Why is the equation not:
x = Asin(t)?

If somebody could clear this up for me it would be greatly appreciated.

Not all SHM has the same period. If we are measuring time in seconds, then ##\sin(t)## would imply a period of ##2\pi## seconds. Whereas, for ##\sin(\omega t)## the period is ##\frac{2\pi}{\omega}##, which covers the general case where ##\omega## determines the period (or vice versa).

For example, if the period of the pendulum is 5 seconds, then ##\omega = \frac{2\pi}{5}##.

If you have ##y=A\cdot \sin(x)## then you have determined a set of certain waves by varying the amplitude. But you can also vary the period, and whether the graph must contain ##(0,0)##. So varying the period gives you ##y=A\cdot \sin(\omega \cdot t)## and a translation out ##(0,0)## an additional constant shift ##C##, so all in all ##y=A\cdot \sin (\omega \cdot t) + C## defines the most general class of waves, if neither amplitude nor period varies in time. We could certainly adjust the coordinate system to make ##A=\omega = 1## and ##C=0##, but what to do if a second wave is considered in parallel?

thebosonbreaker said:
Why is the equation not:
x = Asin(t)?

The argument of the sine function must be radians (or dimensionless).
So, if t has units of time (e.g. sec), then sin(t) doesn't make sense.
Indeed, $\omega$ has units of rad/sec... so that $\omega t$ has units of radians (or is dimensionless).

Side comment:
Similarly, the argument of log() and exp() must be dimensionless.
In the stat mech class I am teaching, I complained about the textbook integrating $\int^{V_{b}}_{V_{a}}\frac{dV}{V}=\ln V_{b} - \ln V_{a} ,$ where $V$ is a volume.
It should be $\int^{V_{b}}_{V_{a}}\frac{dV}{V}=\ln \frac{V_{b}}{V_{a}} .$
If you really want to write a difference then one should write
$\int^{V_{b}}_{V_{a}}\frac{dV}{V}=\ln \frac{V_{b}}{V_{ref}} - \ln \frac{ V_{a}}{V_{ref}} .$, where $V_{ref}$ is any nonzero reference volume.

PeroK, weirdoguy and fresh_42

## 1. Why is the equation for simple harmonic motion represented as sin(wt) instead of sin(t)?

The equation for simple harmonic motion is represented as sin(wt) because it takes into account the angular frequency, w, which is a measure of how fast the motion is oscillating. This allows for a more accurate representation of the motion's behavior compared to using just the regular frequency, which does not account for the acceleration and deceleration of the motion.

## 2. What does the "w" in sin(wt) stand for?

The "w" in sin(wt) represents the angular frequency, which is a measure of how fast the motion is oscillating. It is related to the regular frequency, f, by the equation w = 2πf.

## 3. How does using sin(wt) affect the graph of simple harmonic motion?

Using sin(wt) affects the graph of simple harmonic motion by introducing a phase shift. This means that the motion does not start at its maximum displacement when t = 0, but rather at a different point depending on the value of w. This results in a graph that is shifted to the left or right compared to the graph of sin(t).

## 4. What is the significance of the "w" value in the equation for simple harmonic motion?

The "w" value in the equation for simple harmonic motion is significant because it determines the speed and behavior of the motion. A higher value of w results in a faster oscillation, while a lower value of w results in a slower oscillation. Additionally, the value of w also affects the amplitude of the motion, as seen in the equation A = V/w, where A is the amplitude, V is the maximum velocity, and w is the angular frequency.

## 5. Can simple harmonic motion be represented by any other trigonometric function besides sin(wt)?

Yes, simple harmonic motion can also be represented by the cosine function, cos(wt). The choice between using sin(wt) or cos(wt) depends on the starting point of the motion. If the motion starts at its maximum displacement, sin(wt) should be used, but if it starts at its equilibrium position, cos(wt) should be used.