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High Schoolers rough out a version of Cavendish's Experiment

Faced with students that didn't quite believe Newton's law of gravitation, Anthony Rennekamp
 of Bishop O'Connell High School in Arlington, Virginia, he did what any physicist would do; he had his students build an experiment.  They created a version of Cavendish's experiment using two yard sticks, two one kilogram weights, and a few 50 pound dumbbells.  You can find the write-up of the experiment on the Physics Today website[1].  Mr. Rennekamp and his students also made a video of the experiment.  Given the slow velocities with which the masses move in this experiment, the class reasoned that speeding up a video was the only practical way to view the results.  They published their experimental results video on youtube, (you can watch it below).  There was some online disbelief that the experiment could work as well as it apparently did.  I thought it would be fun to just run a few back-of-the-envelope calculations here and see if their results are reasonable.

The equation for the gravitational attraction between two masses is

$F_g = G\dfrac{m_1m_2}{r^2}$

Where $G = 6.67 \times 10^{-11} m^3/kg\cdot s^2$ is the gravitational constant, $m_1$ is the mass of the first object in kg, $m_2$ is the mass of the second object, and $r$ is the distance between the objects in meters.  As you can see above, as the objects move closer to each other the force due to gravity will increase as $r$ decreases.  Let's ignore that though, and just make an estimate with the force calculated using the initial distance between the two masses, (the mass suspended on the end of the yardstick and the dumbbell).  The distance looks like 7 cm on the video, so we get:

$F_g = 6.67 \times 10^{-11} m^3/kg\cdot s^2 \dfrac{1 kg \cdot 22.68 kg}{0.07m^2} = 3.08 \times 10^{-7} N$

Then, using that force, we can get the time it should take the $1 kg$ mass to travel the $7 cm$:

$a = F/m = 1.51\times 10^{-7} m/s^2$

$distance = 1/2 a t^2$
$t = \sqrt{\dfrac{2*distance}{a}}$
$t = 673 seconds = 11.22 minutes$

While this is only an estimate, (it ignores the dependence of the force on distance; it also ignores the second set of masses), it comes out in the same ballpark as the effect measured by Mr. Rennekamp's class who found that it took 10 minutes for the hanging mass to swing to the dumbbell.  As an extra test, the class placed the dumbbells on the opposite sides of the yardstick and found that the yardstick twisted in the opposite direction as expected.  Pretty Cool!



References:
1.  Write-up of the experiment
http://scitation.aip.org/content/aip/magazine/physicstoday/news/10.1063/PT.5.2025





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