Measuring change in g with height (TV program many years ago)

[CWatters]

Bit of a long shot but can anyone remember a TV program where the presenter measured the weight of something at ground level and compared it to the weight at the top of a tall building, possibly the Empire State building? Sorry I can't recall more details. Think it was at least 10 years ago, possibly more.

 

[hutchphd]

No but I do remember one (a kid's show 3-2-1 contact maybe?) where they were traveling all over Great Britain to find the smallest value of g (it was latitude vs altitude vs nearby mass as I recall)

 

[Chestermiller]

Growing up, I used to watch a wonderful science show on TV entitled Watch Mr. Wizard. it aired in the 1950s, and was presented by a great guy named Don Herbert.

 

[hutchphd]

Upon Don Herbert's death in 2020 the late night host Stephen Colbert paid tribute in the best possible fashion. He performed my favorite Mr Wizard experiment: pulling a hard
boiled egg into a (now old fashioned) milk bottle using a flaming splint of paper to preheat the air. Splint afire, stuff it into bottle, egg on bottle, egg chortles as air expands, splint goes out, egg seals up like fat guy on toilet, kwump..egg in bottle.
I have admired Mr Colbert ever since.

 

[Meir Achuz]

It's better to time 10 (or so) swings of a pendulum.

 

[Mister T]

Bit of a long shot but can anyone remember a TV program where the presenter measured the weight of something at ground level and compared it to the weight at the top of a tall building, possibly the Empire State building? Sorry I can't recall more details. Think it was at least 10 years ago, possibly more.

An episode of Myth Busters. Perhaps.

 

[JT Smith]

I think for a 100m difference in height you'd need a precise way to measure the time it takes a pendulum to swing ten times as the time difference would be on the order of hundreds of microseconds. Using an inexpensive pocket milligram scale would be far easier as the apparent weight would change by hundreds of milligrams for a 1kg mass. The Empire State building is taller than 100m but I think a scale would still be a lot easier to use.

Or maybe I did the math wrong...

 

[hutchphd]

For 300m tall, the change in g should go like $$2x300m/6Mm=100mg/kg$$

 

[JT Smith]

I think you are off by a factor of 10, that it's closer to 1g per kg @300m

 

[Ibix]

$$\begin{eqnarray*}
\Delta g&=&\frac{GM}{r^2}-\frac{GM}{(r+\delta)^2}\\
&=&\frac{GM}{r^2}-\frac{GM}{r^2(1+\delta/r)^2}\\
&=&g(1-(1+\delta/r)^{-2})\\
&\approx&2g\delta/r
\end{eqnarray*}$$So with $\delta=300\mathrm{m}$ and $r=6400\mathrm{km}$ I agree with @hutchphd, about one hundred parts per million for a 300m tall building.

 

[JT Smith]

2 * 9.8 * 300 / 6,400,000 = 0.001 or 1 part in 1000.

What am I doing wrong?

 

[ergospherical]

you want dg/g fam

 

[Ibix]

What am I doing wrong?

Multiplying by $g$. The weight at the bottom and top of the tower is $mg$ and $m(g-\Delta g)$ respectively, so the fractional difference in weights is the difference, $m\Delta g$, divided by the weight at the bottom. That's $\Delta g/g=2\delta/r$.

You've calculated the change in $g$ - put your units in and you'll see it immediately!

 

[JT Smith]

If I want to know the change in weight between the bottom and the top isn't that the difference that I'd be interested in calculating?

 

[Ibix]

If I want to know the change in weight between the bottom and the top isn't that the difference that I'd be interested in calculating?

If so, you need to be consistent with what you are measuring. The decrease in weight of a $1\mathrm{kg}$ mass is indeed $2mg\delta/r\approx 0.001\mathrm{N}$, but the weight is $10\mathrm{N}$ so the ratio is one part in ten thousand. You seem to be comparing the reduction in weight to the mass, which isn't the same thing (or really meaningful at all).

I wasn't kidding about putting your units in. It won't always save you but it will usually flag up when you are doing something daft.

 

[Nugatory]

I wasn't kidding about putting your units in. It won't always save you but it will usually flag up when you are doing something daft.

What he said.

 

[JT Smith]

The OP asked about a show where "the presenter measured the weight of something". He didn't say what the something was so I just took 1kg as a very possible "something" that one might use. For that mass the difference in measured weight on a scale between ground and 300m I believe would be roughly 1g. So for the specific example the units would be grams. But if generalized to any mass it would be grams per kg.

Sorry if I'm still being dense...

If you measured the difference with a scale what would you expect to see?

 

[Ibix]

For that mass the difference in measured weight on a scale between ground and 300m I believe would be roughly 1g.

No, it would be 0.1g, about 1mN.

The change in weight is $mg×2\delta/r$. Work that out for a 1kg mass and you'll get approximately 0.001N. Divide that by $g$ to get the change in (apparent) mass. As far as I can tell you keep leaving out your units and calculating that $mg×2\delta r\approx 0.001$ and assuming that means 1g, not 1mN.

If you measured the difference with a scale what would you expect to see?

I would expect a reading of 10N (or 1kg) at the bottom to reduce to 9.999N (or 0.9999kg) at the top.

 

[JT Smith]

Thanks, now I get it. I was treating the force as if it were grams, probably because it's so common to talk about weight that way. I should have known better but clearly I'm a twit.

It still looks easier to use a scale than a pendulum though. And I guess I'm a little surprised that it's something I could see on my $25 scale. Even a 30m height difference should be detectable. I'll have to try it.

 

[Ibix]

We did the pendulum experiment at university and it turns out there's a 50m height difference across the undergrad lab. That, or undergrads aren't all the most careful experimentalists...

 

[PeterO]

It's better to time 10 (or so) swings of a pendulum.

For noticing a change in g, yes, but to find the actual value you would have to be able to precisely measure the length of the pendulum!.

 

[VACUUMIST]

Hi, look for this paper, specially the von Jolly experiment with a double balance.

Rev. Sci. Instrum. 88, 111101 (2017)