Sunday, April 12, 2009

Let's do the twist

This post is about amateur physics. No computer science is involved, but it was pretty cool.

In 1798 Henry Cavendish did an experiment to weigh the world. Although Cavendish didn't bother deriving the gravitational constant, his experiment was the first that could produce a direct measurement of it.

In the summer of 1985 I was at home convalescing and being bored. It occurred to me one day that if Cavendish could determine the gravitational constant back in 1798, I ought to be able to do something similar, especially because I had access to a few things that were a little hard to obtain in 1798.
The Cavendish experiment involves a torsion balance, which is like a dumbbell which is suspended in the middle by a thin fiber. After letting the thing settle down for a long time, it will be very sensitive to forces that cause it to rotate in the horizontal plane. But any rotation will be opposed (very slightly) by the torsion on the fiber. By placing a pair of large masses near the weights on the end of the dumbbell, the gravitational attraction between the masses will cause the dumbell to rotate until the force of gravity matches the opposing force of the torsion. If we can measure the angle of rotation and determine the torsion of the fiber, we can derive the gravitational attraction between the masses.

Cavendish cast a pair of 1.61 pound lead weights. I found a couple of 2-pound lead cylinders my dad had lying around. I used duct tape to attach them to a 3-foot wooden dowel. Cavendish used a wire to suspend the balance, I used nylon monofilament. To determine the torsion of the fiber, you wait until the balance stops moving (a day or two) and then you slightly perturb it. The balance will slowly oscillate back and forth. The restoring force is calculated from the period of oscillation. Cavendish had a 7-minute period. My balance had a 40 minute period (nylon is nowhere near as stiff as wire).

Cavendish used a pair of 350 pound lead balls to attract the ends of the balance from about 9 inches away. I put a couple of 8 pound jugs of water about an inch away. The next trick was to measure the rotation of the balance. Cavendish had a small telescope to read the Vernier scale on the balance. I used some modern technology. I borrowed a laser from Tom Knight (Thanks again!), and bounced it off a mirror that I mounted on the middle of the balance. This made a small red dot on the wall about 20 feet away. (I was hoping this would be enough to measure the displacement, but I was considering an interferometer if necessary.)

To my surprise, it all worked. After carefully putting the jugs of water in place, the dot on the wall started to visibly move. Within a few minutes, it had moved an inch or two. I carefully removed the jugs of water and sure enough, the dot on the wall drifted back to its starting position.

This was really cool. Newton's theory of gravity was the first ‘unified theory’ of physics. It took several disparate phenomena — the orbits of the planets, the orbits of moons, tides, and the kinematics of falling objects — and proposed a single theory that explained them all mathematically. But Newton's theory supposed that every object has a slight gravitational attraction to every other. This is a strange phenomenon that hadn't been observed (prior to Cavendish). It's not something you usually see because the force is extraordinarily small.

My dad was astounded. Of course he knew about gravity from high school, and knew it kept the planets in orbit and stuff stuck on the ground, but he hadn't remembered (or perhaps wasn't taught) that there was a very small gravitational force between everything, including the lump of lead on the end of my stick and the jug of water a few inches away.

Even though I knew there was a force I could measure, it was still pretty amazing to watch it happen. Sure, you believe Newton's laws, but after seeing this in action, there is still a `wow' factor.

Now as for the value of G. I think I gave enough information here for someone to derive it. I think I calculated it to be somewhere around 10^-11 plus or minus an order of magnitude. One day I might try to really calculate it.

I have to recommend trying this experiment if you have the room to set it up. It's something to see.


downthehillside said...

Wow, that's pretty cool. It's kind of weird that it took 120 years for someone to do an experiment like that - the Principia was published in 1687. I wonder if people had tried something similar and failed to get it to work. Seems like it could have been done with rope.

I thought torsion clocks had been around longer, but Wikipedia says the torsion pendulum was only invented in 1793, and the clock in 1841.

pbhj said...

Could use a diagram! Over than that quite interesting. When my lad is a bit older this is definitely one we'll have to do.

Lincoln Quirk said...

Whew! It actually came out kinda close. Based on my assumptions from your description

G = 1.32 * 10^-10

whereas Google says

G = 6.67300 × 10^-11 m^3 kg^-1 s^-2

(too big by about a factor of 2)

Here's my math:


F = ma (Newton's motion)
x = sin t/(2pi*T) (simple harmonic motion with period T)
F_s = -kx (Hooke's law)
F_g = G m1 m2 / rad^2 (newton's grav.)

And observations

m1 = 0.9kg
m2 = 3.6kg
T (period of oscillation) = 40 min

To find the distance between centers of mass, for the G calculation, I had to guess based on the objects you used.

You said 1 inch, I assumed this was one inch separation between edge of the water jug and edge of the lead cylinder. Assumed radius of milk jug = 9cm; lead cylinder = 2.5 cm; separation = 2.5cm, for a total of 14cm between center of mass.

rad_g = 0.14m

I also had to assume you placed the centers of the lead cylinders at the end of your 3ft dowel.

rad_dowel = 1.5ft

And I also assumed that you were projecting the laser onto a perpendicular wall (20ft away):

rad_wall = 20ft

light moved "1 or 2 inches": I assumed 3cm

delta_x_wall = 0.03m


First, find the spring constant, k, for your hanging system.

Doubly differentiating the harmonic motion equation and solving leads to:

a = -x / (2pi * T)^2

Substitute f = kx and f = ma

k = m / (2pi * T)^2

Use m = 2m1 (2 lead weights in motion). Neglect the mass of the dowel.

I got k = 1.9 * 10^-5 kg/sec^2

Now we can figure out G.

By similarity of triangles,
delta_x = delta_x_wall * rad_dowel / rad_wall

By hooke's law, F_spring = k (delta_x). Since we are concerned with the system's new rest position, total forces are zero so F_gravity = F_spring.

There are two instances of gravitational interactions we're measuring which sum:

F_spring = F_gravity
k*(delta_x) = 2 * G * m1 * m2 / rad^2

Plug in values and solve for G.

Nathan said...

I wish you'd taped it!

fuf2 said...

how can you be sure electrical attraction is not interfering? I understand those forces are much stronger and might have contributed to the balance moving.

Geoff Knauth said...

This is really neat. I'm going to share this with my kids (12 and 10).