Have what you need to calculate the phase shift. Next measure the period of the motion, and you In the same direction as the red ball was moving at that time. Play the animation from t = 0 to the point where the blueīall reaches the initial position of the red ball and with a velocity There's another way to determine the phase shift using theĪnimation. That the two graphs coincide with each other very nearly. Now that we have the time shift and the period,Įnter that value into the input box for the Phase of Blue. Quite 2 grid divisions less than the scale maximum of 3.0 s. So we can read the time shift as 5.1 grid divisions orĠ.51 s. Graphs cross x = 0 going downward.) Note at the bottom that the time (WeĬould also have selected the two earlier points where the red and blue The time shift is the difference in time between marksĪ and b where the red and blue graphs cross x = 0 going upward. The display should be something like the following (without the Read the description. Then run the animation time, then we canįor an example of how this is done, open this animation. If we canĭetermine t s from a plot of position vs. Where t s is to be interpreted as a time shift. In like manner, we substitute for the phase shift, The following sequence of equations shows how this arises. You may find it easier to interpret the phase shift as a time shift. We call the phase shift, then we write position as:, where we're assuming an equilibrium shift of 0. However, weĬan express position more generally using a phase shift. In section A, the phase of the position was 0.
Simple harmonic motion equations how to#
We'll now go on to see how to deal with phase shifts. The result is the same as the precision of the vertical scale and is therefore attributable to sighting error in reading the scale. At t = 1.6s, y = 0, according to the graph. Putting it all together, the equation of the position vs.
![simple harmonic motion equations simple harmonic motion equations](https://i.ytimg.com/vi/gJglfKEi-K0/hqdefault.jpg)
The period can be determined from the horizontal difference between 2 peaks. The amplitude of the motion is half the difference of the maximum and minimum positions:Ī = /2 = 0.235m, or 0.24 to two decimal digits. The easiest way to determine the value of equilibrium shift from the graph is to average the maximum and minimum positions. The equation of the graph above is y = Acos( ωt) + y eq, where y eq is interpreted as a vertical shift due to the fact that the equilibrium position is not y = 0.
![simple harmonic motion equations simple harmonic motion equations](https://i.ytimg.com/vi/c24M8m5SJRc/maxresdefault.jpg)
A screen capture of the graph is shown below. Click on Show Graph to see the position vs. Run the animation and note that the vertical path of the oscillation isn't centered on y = 0.
![simple harmonic motion equations simple harmonic motion equations](https://image.slidesharecdn.com/simpleharmonicmotion-100703004526-phpapp02/95/simple-harmonic-motion-10-728.jpg)
In real-world situations, such as a mass oscillating vertically on a spring, the equilibrium position of the system may not correspond to a vertical position of 0 on a graph. Equation of motion in the case of an equilibrium shift We continue below the graphs and write the equationsī. We can see that the latter two values agree with theĬorresponding graphs. We first obtain the following information Of a cycle before that (or a half cycle before the position.) Let's Quarter of a cycle before the position and the acceleration peaks a quarter The graphs below show position, velocity, and accelerationĪll plotted on the same time scale. In phase by by radians (180° or a half cycle) from position. Now let's put the equations one below the other.īy expressing position, velocity, and acceleration asĬosine functions, we see that velocity is shifted in phase by radians (90° or a quarter cycle) from position and acceleration is shifted The equation for acceleration can be expressed as follows: The equation for velocity can be expressed as follows: Of the quantities are simply those for which or -1. See Section 13-2 of the text for more discussion ofįor an object oscillating in SHM with angular frequency and released from rest at a position x = A, the position, velocity,Īnd acceleration as a function of time are:įrom these equations, one can see that the maximum values Phase relationships between position, velocity, and acceleration for an object in simple harmonic motion
![simple harmonic motion equations simple harmonic motion equations](https://slideplayer.com/16/4919369/big_thumb.jpg)
Equations of Simple Harmonic Motionĭownload this Excel file in order to experiment with changing the various parameters in order to see how that influences the graphs of position, velocity, and acceleration vs.