Back in the wonderland of physics that is your garage, you and your friend decide to
construct a simple pendulum.
This string has a length of two meters, and the ball at the end has a mass of 0.5 kilograms.
You then try to make various predictions about the system.
If you release the pendulum from an angle of thirty-five degrees below the horizontal,
where will the ball be traveling the fastest, and what will its velocity be at that point?
To answer this, we need to know about potential and kinetic energies, as well as some trigonometry,
so check out those tutorials first if you need to, otherwise go for it.
Just as with an object in free fall, this pendulum will always have some amount of potential
energy, and some amount of kinetic energy.
Just before releasing, it has only potential energy, as kinetic energy is the energy of
motion, and it isn’t moving.
Once released, the potential energy is converted into kinetic energy, until it reaches the
lowest point, where there is zero potential energy and only kinetic energy, and then back
up to the other side, converting all the kinetic energy back into potential energy as it slows
down, until for a single instant it has zero kinetic energy once again.
This clearly means that the point where the ball will be moving the fastest is right at
the bottom, where kinetic energy is at a maximum.
And specifically, this amount of kinetic energy will be equal to the initial potential energy,
so let’s calculate that first.
This will be equal to MGH, and we can take the bottom of the swing as being zero height.
So what is the height at the point of release?
To find that, we need to construct a triangle.
We have this angle, thirty-five degrees, and we have the hypotenuse, as the string length
is always equal to two meters.
We want to know this opposite side, which we can get by using sine.
Sine theta is opposite over hypotenuse, so sine of thirty-five equals this distance X
over two.
We evaluate the sine, multiply by two, and we get 1.15 meters.
Now that we have this distance, it is easy to get the height from the lowest point, because
when vertical, this length is two meters, so this length must be two meters minus 1.15
meters, or 0.85 meters.
So there is our H, and we have M and G. 0.5 kilograms, 9.8 meters per second squared,
and 0.85 meters, that will give us 4.17 joules.
Now as we said, the potential energy at this point, will equal the kinetic energy at this
point, and kinetic energy equals one half MV squared.
So once here, 4.17 joules equals one half the mass times velocity squared.
So we multiply by two, divide by the mass, and then take the square root, and we should
get 4.08 meters per second.
Plenty fast enough to scare your cat, or whatever else you weirdos intended to do with it.
Thanks to physics!