the world in a different light
Here’s my attempt at animating 5 graphs for simple harmonic motion together in one page.
From left column:
$$v = \pm\omega\sqrt{x_o^2-x^2}$$
$$a = -\omega^2x$$
From right column:
$$s = x_o\sin(\omega t)$$
$$v = x_o\omega \cos(\omega t)$$
$$a = -x_o\omega^2 \sin(\omega t)$$
And here is the animated gif file for powerpoint users:
The first of two apps on Phase Difference allows for interaction to demonstrate the oscillation of two different particles along the same wave with a variable phase difference.
The second shows two waves also with a phase difference.
In both cases, the phase difference $\Delta\phi$ can be calculated with
$$\Delta\phi = \dfrac{\Delta x}{\lambda} \times 2\pi$$
where $\Delta x$ is the horizontal distance between the two particles or the horizontal distance between the two adjacent identical particles (one from each wave) and $\lambda$ is the wavelength of the waves.
I modified Tom Walsh’s original GeoGebra app to add a moveable single oscillating particle for students to observe its movement along a longitudinal wave and a transverse wave.
The app can also be used to show how the displacement of a particle in a longitudinal wave can be mapped onto a sinusoidal function, similar to the shape of a transverse wave. For example. a displacement of the particle to the right can be represented by a positive displacement value on the displacement-distance graph.
You can choose to select the particle that you want to focus on by using the slider.
For a full screen view, visit https://www.geogebra.org/m/auyft2pd
Here is an animated gif for those who prefer to insert it into a powerpoint slideshow instead:
For embedding into SLS or any platform that supports iframes.
<iframe scrolling="no" title="Progressive Waves" src="https://www.geogebra.org/material/iframe/id/auyft2pd/width/640/height/480/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/false/ctl/false" width="640px" height="480px" style="border:0px;"> </iframe>This GeoGebra app allows students to observe closely the movement of a particle in a progressive wave, with two possible directions of energy propagation.
In a typical question, students will be asked to predict the next movement of a particle given that a wave is moving left or right. Usually, students will need to imagine the waveform shifting slightly to the left or right in order to figure that out. This app follows the same visualisation technique to identify the subsequent movement of any particle along a wave.
Using a chain of rubber bands, I swung a ball around in a vertical loop. This demonstration shows how the tension in an elastic band changes according to the position of the ball, by referring to the length of the elastic band.
When the ball of mass $m$ is at the bottom of the loop, the centripetal force is given by the difference between tension $T_{bottom}$ and weight $mg$, where $T_{bottom}$ varies depending on the speed of the ball $v_{bottom}$ and the radius of the curvature $r_{bottom}$.
$T_{bottom} – mg = \dfrac{mv_{bottom}^2}{r_{bottom}}$
When the ball is at the top of the path, it is given by
$T_{top} + mg = \dfrac{mv_{top}^2}{r_{top}}$
As the weight is acting in the same direction to tension when the ball is at the top, a smaller tension is exerted by the elastic band to maintain a centripetal force. Therefore , $T_{bottom} > T_{top}$.
The GeoGebra app below shows a simpler version of a vertical loop – a circular path with a fixed radius $r$. Consider a ball sliding around a smooth circular loop. The normal contact force varies such that
$N_{bottom} = \dfrac{mv_{bottom}^2}{r} + mg$
$N_{top} = \dfrac{mv_{top}^2}{r} – mg$
It can be shown that the minimum height at which the ball must be released in order for it to complete the loop without losing contact with the track is 2.5 times the radius of the frictionless circular track.
If we were to consider the rotational kinetic energy required for the ball to roll, the required initial height will have to be 2.7 times the radius, as shown in the video below:
Many thanks to Dr Darren Tan for his input. Do check out his EJSS simulation of a mass-spring motion in a vertical plane, which comes with many more features such as the ability to vary the initial velocity of the mass, graphs showing the variation of energy and velocity, as well as an option for a mass-string motion as well.