[This a short version of the earlier Finnish blog post, read the original if you want to understand more, of course learn Finnish first.]
I just finished the Galilei biography (Heilbron J.L. Oxford, 2010). A really nice book, it took about half year to read it. Reading it reminded me about the theorems on inclined planes. I have been studying a long time Galilei’s Dialogues books in English. Still most of the geometric proofs are very hard to understand because my intelligence on Euclidian geometry is not very good. In his Discorsi e Dimostrazioni Matematiche, intorno a due nuoue scienze he states in Third Day Theorem VI, Proposition VI ”If from the highest or lowest point in a vertical circle there be drawn any inclined planes meeting the circumference the times of descent along these chords are each equal to the other.”
If you are familiar with Newtonian mechanics, it is not very hard to prove this :o)
Earlier in the same Third Day in his book he had already shown that the space travelled on constant acceleration is proportional to acceleration and time squared s = ½ a t2. And the acceleration on inclined planes without friction in constant.
We actually have two theorems.
- Start from the the top of the circle with particles sliding without friction with different angles. At the same time they should be on the same circle.
- Start from the circumference of the circle so that the planes meet at the lowest point, they should meet at the same time.
But wait a minute, what happens after that. That was the thing that I was wondering when I started this study.
Lets study this with GeoGebra.
Everybody who loves physics can see that the acceleration on inclined plane without friction is a = g sin alpha , where g is the gravitational acceleration and alpha the horizontal angle. I will forget g and ½ in acceleration because Galilei’s theorem works on the moons of Jupiter also.
[Just noticed that my alpha was changed to a in WordPress, so after this a in the angle.]
In this 1st case it easier to use polar co-ordinates. In GeoGebra you can create a point in polar co-ordinates, if you use ; as a separator. If you want to create a point from origin with length 2 and angle 20°, the write to Input
XXX = (2; 20°)
You should see the point in Graphics window. This was just a test, so delete XXX or just ignore it.
For us the co-ordinates will be (sin(a°) t²; (-a)°). Think why the other has the negative sign.
Let’s create slider for time t. Write to the Input
Toggle button t and click it with mouse right button and choose object properties Min:0, Max: 5, Increment: 0.1
Write to the Input:
Sequence((sin(a°) t²; (-a)°), a, 0, 180, 10)
If you move the slider t, you will see that the points seem to move on the same circle.
Beautiful. Prove it.
See it on GeoGebra Materials.
Wait for the next story, it even more beautiful.
I am going fishing.