[TomFoolery]

So since my previous experiment, while apparently detecting gravity type forces, did not yield a meaningful measurement of those forces:
It was close enough to the value of gravity that my rinky dink experiment may have caused that much error, but far enough that it wasn't even close.

So I've begun efforts to actually measure it. I'm intending to get a measurement accurate enough to either match theoretical gravity, or rule it out.

My setup is a roughly 70 gram tungsten weight suspended from two strings with a small fragment of mirror attached. The weight is hanging in water to slow down the swinging.

Then a laser interferometer setup bounces 532nm laser light off the mirror on the weight and the fixed reference mirror and the combined beams hit some white paper to form an interference pattern.

Theoretically, I will be able to measure movement of the weight in increments of half of 532nm since moving the weight 1nm increases the distance by 2nm.

And depending on how far the interference bands move I may be able to even guesstimate to the nearest quarter of 532nm.


It's still taking forever to settle.

Shown first  is a test setup with a red laser.
Second, I'm showing it actually set up and "operating" with green laser, except it's still swinging too much, but the interference pattern shows up briefly at each end of the swing as the velocity briefly hits zero.
Sorry about the poor focus, my phone does not like focusing on green laser light. It's probably looking for face colors to focus on, and green is not that.
Third is showing the interference patterns as the weight still swings after an hour (while swinging in a cup of water.)

(The interference pattern is actually still there, but moving so fast it's a blur)

I could actually set up a pair of photo diodes at 90 degrees phase difference and track the quadrature and measure the swinging in realtime, but that's too much work to do before going out to dinner with friends.

Looks like this thing may never settle down completely, and even if it does, when I walk around to move the big weight nearby, the floor moves and the pattern goes crazy.
I guess 532nm isn't a very big distance.

[TomFoolery]

So I got the weight to stop swinging. It's hanging down into a tuna can which has water (for anti-static electrical connection) and mineral oil because of it's higher viscosity.
It now stops swinging pretty quickly.

However, it seems there is constant vibrations that prevent a stable reading.

Even just standing very still, it's not stable. But if I move or take a step, it shakes the whole house on the nanometer scale.

So unfortunately the laser interferometer method will have to wait until I can do it on a concrete floor. I just hope that is more stable!

[WellRoundedIndividual]

Do you have access to a slab of granite? The type the use for CMMs? And then some vibration dampener? Maybe some motor mount blocks?

[TomFoolery]

Do you have access to a slab of granite? The type the use for CMMs? And then some vibration dampener? Maybe some motor mount blocks?

I don't have access to a slab of granite, but I do have a 4x6ft by 1in thick pane of glass, which I suppose would work. I don't have a place to set it up but Lord willing in a month or two I will.

I could definitely rig up something rubbery for absorbing vibrations.

Thanks for the idea!

[WellRoundedIndividual]

There are plenty of DIY examples of people making their own vibration isolation tables.  Not sure of the effectiveness but the idea is simple enough.

[TomFoolery]

Since I can't set up a vibration damped table setup, I decided to try the other idea.

I have a small tungsten weight suspended in a vacuum on two fishline strings.

There's a phototransistor taped to one side of the jar, and a small laser diode with the lens removed which is taped to the other side of the jar.

The weight casts a swinging shadow on the phototransistor.

A Microchip measures the average time and swing count and stuff from the phototransistor.
It also sends little pulses at zero crossing to the coils which are taped on the jar to keep the weight swinging.

It also has a small magnet on it so the electromagnet coils (air core) can act upon it.

The idea is to get an average swing rate over one or more days, then put the lead blocks under it, and repeat the measurement, and see if I can't derive the actual force being caused by the lead weights.

The reasoning being that since the force is very weak, I can let this thing run for a long time to increase my signal to noise ratio and my accuracy.
Since my CPU has a resolution of about 1/24000000th of a second, if I average for a day that brings my granularity up to one part in 2 trillion.
That ought to be enough to resolve the gravitational differences to at least a couple significant digits.

But we'll see.

[TomFoolery]

Just an update, the apparatus I think is ready for utilization.
It's set up where I don't bump it's table as often, and it's running, gathering an average swing period of the pendulum.

It reports the period (with microsecond accuracy,) every period, and reports a sum time for 250 swings in 24ths of a microsecond.
(It takes about 250 swings to get near the 4 billion limit of an unsigned 32 bit integer.)

It's set up so there is a place under it where I can slip in my lead bricks to see if that changes the frequency.

It's been running for about 12 hours, and it does seem to still be affected when people walk around the house or open/shut the door.
However, the reading for 250-cycle averages is pretty stable, usually staying within a +/-10ppm region, although sometimes going outside that but within a +/-25ppm region.

My plans are to run it for a day without the lead weights under it then a day with, and see if I see a difference of significant digits.
If not then a week and a week.

[QED]

Tom,

Cool idea. But why not place the lead weights on one side?

When beneath, then any effect from the weights would reduced by drag forces - they both would decrease amplitude. On the side they may partially cancel. I see that you pumped the container,  but could not read the pressure. What is your estimate of the maximum drag force on this system (I.e, drag force at v_max)?

[TomFoolery]

Tom,

Cool idea. But why not place the lead weights on one side?

Because that wouldn't change the frequency, it would only change the local level, and only very slightly. My setup is not ridged enough to detect a change in local level reliably.
(However, my firmware does measure the local level based on how much time the pendulum spends on each side of the zero crossing sensor, so theoretically I could detect a change in local level.)
But for now, I just want to measure change in frequency.

When beneath, then any effect from the weights would reduced by drag forces - they both would decrease amplitude. On the side they may partially cancel.

I'm not sure I follow what you're trying to express there, my friend. Perhaps try again?
More gravitational pull downwards would increase the swing frequency.
The swing frequency is a function of two things: Pendulum length, and gravitational force.

I see that you pumped the container,  but could not read the pressure.

The gauge was a 15psig fuel pressure gauge which I modified to be a vacuum gauge. In free air, it reads about 15psi. Under full vacuum, it reads 0 PSI. roughly. It's just a fuel gauge, but it lets me check to make sure my jar hasn't sprung a leak.
It's reading around a PSI or a half a PSI, but pressure is probably less than that. About as good as my cheapo HVAC vacuum pump can get.

What is your estimate of the maximum drag force on this system (I.e, drag force at v_max)?

No clue.
But it takes extremely little positive feedback to keep it swinging nicely.
I have not tested yet how long it swings without positive feedback.

[QED]

Right, and local gravity would be affected with weights on one side. You have a tension, and weight, and now a grav force from the lead side weights. Those three vectors add, and so this would certainly impact the frequency.

The direction of oscillation has a sine component of the weight and a component in the opposite direction from the lead weights. Just draw a free body diagram and write Newton’s laws to produce the wave equation. Take sine_theta ~ theta for small angles.

The drag should be quadratic in speed and bases on a other coefficients.

D~ (1/2)*rho*A*v^2, where tho is the fluid density, and A is the cross sectional area. In reality you should include a form factor, but it is of order unity so not that important.

The main thing behind my query was to check that the drag forces were not of the same order as the force you are trying to measure. It is a good thing to check.

[TomFoolery]

Right, and local gravity would be affected with weights on one side. You have a tension, and weight, and now a grav force from the lead side weights. Those three vectors add, and so this would certainly impact the frequency.

Hmm, I'm not sure about that. If I add gravitational pull from above, it'll reduce the frequency. If I put gravitational pull from below, it'll increase the frequency.
However is there not a position on the side where if I add the pull of gravity it will neither increase nor decrease the frequency, but simply cause a slight tilt in the apparent local level?
I'll have to think about that.

The direction of oscillation has a sine component of the weight and a component in the opposite direction from the lead weights. Just draw a free body diagram and write Newton’s laws to produce the wave equation. Take sine_theta ~ theta for small angles.

The drag should be quadratic in speed and bases on a other coefficients.

D~ (1/2)*rho*A*v^2, where tho is the fluid density, and A is the cross sectional area. In reality you should include a form factor, but it is of order unity so not that important.

The main thing behind my query was to check that the drag forces were not of the same order as the force you are trying to measure. It is a good thing to check.

Are you talking about the drag forces of air resistance and bearing resistance which reduce the Q, or dampen the oscillation movement?

That's why I put it in a vacuum, to reduce air drag.
And that's why I used a razor blade edge pivot.
It feels like we're talking different languages, I hope I'm understanding you correctly!

[QED]

Yes, it sounds like you come at this from an engineering background, whereas I come from a physics one. So our way of thinking about it may differ. No matter, I’ll just be more careful in my exposition.

So the equation for a simple harmonic oscillator derived for a pendulum is a second order differential equation with angular frequency given by /sqrt(g/L). If you add another force on the side, this will modify this equation to yield an angular frequency of /sqrt([g-g’]/L), where g’ is the acceleration from the lead weight: GM/d^2.

For the drag, yes you pumped the jar, but it is not a pure vacuum in there, right? You just reduced the air density beyond the precision of your psi meter. Achieving a near vacuum is an expensive and time consuming process requiring heat pumps, throttling techniques, and some others I do not recall at the moment. So your drag is reduced, but not absent. Since the force you attempt to measure is very small, are you certain that the drag has been reduced enough? It could be that the drag is still the same order of magnitude as the effect you purpose to measure. That’s why checking this is useful.

[TomFoolery]

Yes, it sounds like you come at this from an engineering background, whereas I come from a physics one. So our way of thinking about it may differ. No matter, I’ll just be more careful in my exposition.

So the equation for a simple harmonic oscillator derived for a pendulum is a second order differential equation with angular frequency given by /sqrt(g/L). If you add another force on the side, this will modify this equation to yield an angular frequency of /sqrt([g-g’]/L), where g’ is the acceleration from the lead weight: GM/d^2.

Question:

Do you agree that if I put more mass above my pendulum it will slow down, and if I put more mass below it, it will speed up, and that at the correct position besides it, more mass would neither slow it nor speed it?

For the drag, yes you pumped the jar, but it is not a pure vacuum in there, right?

Yeah, probably a 99% vacuum. My micron vacuum gauge is on loan to a friend at the moment, but my HVAC pump does a pretty good job.
Good enough that a goose down feather falls like a BB and bounces in the vacuum. So I would say the drag on my round dense tungsten weight is probably quite minimal.

You just reduced the air density beyond the precision of your psi meter. Achieving a near vacuum is an expensive and time consuming process requiring heat pumps, throttling techniques, and some others I do not recall at the moment. So your drag is reduced, but not absent. Since the force you attempt to measure is very small, are you certain that the drag has been reduced enough? It could be that the drag is still the same order of magnitude as the effect you purpose to measure. That’s why checking this is useful.

You may be missing the entire underlying principle of my experiment.

Drag primarily causes loss of efficiency. The pendulum just won't swing as long on it's own if there's too much drag. But the frequency will still be rather stable because it's primarily determined by the length of the pendulum and the force acting upon it.
So if I can maintain constant drag (by having it in a vacuum jar) and constant pendulum length, then the only other major variable is the gravitational force in the vertical axis.

Now of course there are other small effects, temperature can change the length of my aluminum pendulum shaft, light can create a photoelectric effect and charge up my pendulum, and so on and so on.
And a shift in local level (like if the table it's on gets tipped a bit) might upset the frequency a little due to the fact that the moment at which the excitation pulse is fired will actually not be centered in the swing.

But my plans are, Lord Willing, to actually move it to a place where it's bolted down to a 2000 pound steel machine that's sitting on concrete, and put a light shield around it to reduce possible outside influences.

But anyway, even if there is drag, the frequency is still primarily a function of pendulum length and gravitational force in the vertical axis.
So by taking a long reading without the lead weights under it then another long reading with the lead weights under it, without changing the drag or anything else, then I should be able to get a difference in frequency and from that calculate the difference in gravitational force.

Does that make sense?

Can you see how the effect of the drag would be self-canceling, and it's really a differential measurement where gravitational force is the only major input variable?

And if you have a better idea for measuring gravity I'm all ears!

[QED]

The drag force is a function of speed, and will definitely not be constant, because the pendulum is undergoing continual acceleration in its trajectory. So it will affect the frequency. Like I said: solve the equations, you will see this.

I like your method, and am simply pointing out a concern you may want to check before interpreting your results.

This is why we make our engineers take physics courses

[TomFoolery]

The drag force is a function of speed, and will definitely not be constant, because the pendulum is undergoing continual acceleration in its trajectory. So it will affect the frequency. Like I said: solve the equations, you will see this.

I like your method, and am simply pointing out a concern you may want to check before interpreting your results.

This is why we make our engineers take physics courses

It's just too bad the physicists don't have to take engineering courses 

But I really look forward to seeing some of your experiments!

But seriously, just for you, I released the vacuum. Opened the bottle to atmospheric pressure. And let it run for several hours.
The pendulum is definitely not swinging as far, but the frequency did not deviate outside of it's normal variation over the past two days.

Regarding the graph below:

Vertical scale is Hz.
Horizontal scale is measurement count. Each measurement is an average of 250 swings.

[QED]

The drag force is a function of speed, and will definitely not be constant, because the pendulum is undergoing continual acceleration in its trajectory. So it will affect the frequency. Like I said: solve the equations, you will see this.

I like your method, and am simply pointing out a concern you may want to check before interpreting your results.

This is why we make our engineers take physics courses

It's just too bad the physicists don't have to take engineering courses 

But I really look forward to seeing some of your experiments!

But seriously, just for you, I released the vacuum. Opened the bottle to atmospheric pressure. And let it run for several hours.
The pendulum is definitely not swinging as far, but the frequency did not deviate outside of it's normal variation over the past two days.

Regarding the graph below:

Vertical scale is Hz.
Horizontal scale is measurement count. Each measurement is an average of 250 swings.

Hi Tom!

I am a theorist, and so avoid experimental efforts wherever possible

One cannot really draw conclusions based on this data - at least not whether the drag force impacted the frequency. What you would need to do is a statistical test to see if there is any significant difference. Perform a regression on each set, and compute a hypothesis test to see if the y intercepts are different at some confidence level. You’ll have to be careful, because you have a different number of trials for each set, which modifies how well you know the intercept. A proper reduced chi squared regression will account for this, of course.

One does not simply visually inspect a data set to form a scientific conclusion.

[TomFoolery]

Well, interesting but meaningless results after 3 days.

For clarification, days start around 9AM to 11AM.

After the first day, I inserted 23 pounds of lead weights.
After the second day I released the vacuum.

Neither had a significant effect compared to the other variations.
I have not yet figured out the cause of the other variations.

I think I'm going to have to run this thing for a month to see if maybe I'm picking up sun and moon gravitational pull.

Anyway, here's what I got so far.  If anyone wants the raw data I can send send it along.

 

[QED]

Thanks!

It seems like there is quite a change when you release the vacuum on the frequency, yes?

[TomFoolery]

Thanks!

It seems like there is quite a change when you release the vacuum on the frequency, yes?

Yes and no.

There was constant change for the whole 3 days.

The frequency was in the middle of a downward trend when I released the vacuum.
Me touching the jar caused that spike where it says "released vacuum" but you can see that it continued on the exact same trend after that and continued to drop.
Then after several hours, it started going back up and eventually did set a new high for the 3 day period.

So if the air has an affect on the frequency, it increases the frequency, which makes no sense, the air should slow the frequency.
But the overall frequency during the air-filled day was higher than either of the previous two days.
And the trend of frequency change was not apparently affected by the loss of vacuum.

My guess is the variation we see is from something other than the vacuum or air-drag.

I'm really not sure what is going on, although I'm wondering if it has to do with the sun and moon and stuff.
Maybe even the mass of clouds? Beats me.

But I do hope to do a month long test so I can correlate to the sun/moon, local tides, and whatever.

[QED]

There could be non-isotopic local currents that drive the system. This could increase the frequency.

Which one would be higher, the gravitational force on your system from the Sun, or from YOU?