76
votes
Why should a clock be "accurate"?
why it is necessary for a reference clock to worry about this, if the definition of the second itself is a function of the number of ticks the clock makes.
The concern is that somebody else (say a ...
- 23.2k
42
votes
Why should a clock be "accurate"?
For most of human history, we had a single mechanical clock: the spinning Earth.
Well, actually two mechanical clocks. The Earth’s spin rate is a good constant, but it’s tricky to measure directly. ...
rob♦
- 74.5k
36
votes
How long is a second?
A second is a second long by definition, but if you measure any time in seconds, the number of seconds you infer will be subject to an error of at least $\mathcal O(10^{-15})$ because of the ...
- 4,472
25
votes
Accepted
The Electron at the End of the Universe
The calculations are done in
Schwartz (2019), Lecture 3: Equilibrium (https://scholar.harvard.edu/files/schwartz/files/3-equilibrium.pdf)
Here's a summary of the key equations, using the notation ...
- 51.4k
23
votes
If a measurement has 5% error, can we say it has 95% accuracy?
Prefer “uncertainty” over “error.” When you say “error” you imply that Someone Out There has determined the Right Answer. This isn’t how it works outside of an introductory lab class.
When you say “...
rob♦
- 74.5k
23
votes
Why should a clock be "accurate"?
if the definition of the second itself is a function of the number of ticks the clock makes.
Why don't we just use a single simple mechanical clock somewhere with a wound up spring that makes it tick,...
- 7,745
15
votes
Why should a clock be "accurate"?
Having a central clock system has a lot of drawbacks:
broadcasting means the signal takes time, so if I need a clock, it will always be somewhat behind. This cannot be fixed to the precision ...
- 15k
15
votes
What if the resulted error is so large?
This is a case in which the standard statistical tool of Gaussian error propagation fails.
When you write a measured quantity as "$\text{quantity} = \text{value} \pm \text{error}$" this is ...
- 4,744
13
votes
Accepted
Rules of significant figures, precision, and uncertainty
There is actually no guarantee at all.
Although in school students are taught about “significant figures”, they are not used by professional scientists. They are like “training wheels” for ...
- 69.2k
11
votes
What if the resulted error is so large?
$F = \frac{1}{A-B}$
with $A=1.08\pm0.02$ and $B = 1.05\pm0.03$.
...which looks ridiculous. So, is this error analysis problematic?
It's not ridiculous. It might be problematic depending on how well ...
- 8,593
7
votes
How long is a second?
The second itself does have an uncertainty. When we're using it without uncertainty, we're basically using the following trick:
The time span of $n$ seconds is defined as $2\times n\times 9\,192\,631\...
- 14.1k
7
votes
Can we measure $10^{-12}\ \mathrm{N}$ force?
The question is, a $10^{-12}\rm\,N$ force applied to what. A force of $10^{-12}\rm\,N$ applied to a hydrogen atom, with mass $10^{-27}\rm\,kg$, would produce an acceleration $F/m = 10^{+15}\rm\,m/s^2$....
rob♦
- 74.5k
7
votes
Uncertainty notation: I am unsure of how the parentheses notation works
In the first place, I would write
$$ 5.868\,709\cdots×10^{−7} \pm 7.884\,31\cdots×10^{−12} $$
instead as
$$ (5.868\,709 \pm 0.000\,078\,8431 )×10^{−7} $$
with (a) grouped digits, (b) a common exponent,...
rob♦
- 74.5k
7
votes
Why should a clock be "accurate"?
Time doesn't flow, nor is it perceived, according to the ticking of a clock. If you boil an egg while watching a clock that runs slow, you're going to overcook it, regardless of the fact that the ...
- 6,730
7
votes
Why should a clock be "accurate"?
I am adding this answer because I feel like the other answer do not cover some main parts as to really why we a destined to create more and more accurate clocks: more accurate distance calculations ...
- 25.9k
7
votes
Multiplication and significant figures
I like to teach my students to trust their hunches about significant figures using their emotional response to the rounded number. I call this the “anger management method of error estimation.”
When I ...
rob♦
- 74.5k
6
votes
How long is a second?
There's a shift from the old way of having standard examples of the units that everyone can compare against, to defining the units in terms of fundamental physics.
There was an uncertainty in ...
- 5,097
6
votes
Accepted
What is the concise form of physical constants? (What is the number in the parentheses?)
The numbers inside the parentheses represent the uncertainty.
If you wanted to explicitly include the uncertainty in the proton's mass, you would write it as $$m_p=(1.672\ 621\ 923\ 69 \pm 0.000\ 000\...
- 25.6k
6
votes
Rules of significant figures, precision, and uncertainty
It depends on the device. You raise two separate but related issues.
For an explicitly digital device, a common misconception is that the uncertainty can be as small as half of the least significant ...
rob♦
- 74.5k
5
votes
Calculating uncertainty from significant figures of a value
Most likely, the authors assume Gauss error propagation in which the error on a function $f(x)$ of a variable $x$ is calculated as $$ \Delta f = \frac{\partial f}{\partial x} \Delta x~.$$ In your case,...
- 592
5
votes
Is an ideal Cesium Standard atomic clock more accurate than Einstein's thought experiment light-clock?
If we are considering real clocks then the caesium clock wins hands down.
A second is defined as the time taken for 9192631770 oscillations at the frequency of the caesium hyperfine line. So once you'...
- 334k
4
votes
What is the concise form of physical constants? (What is the number in the parentheses?)
It's the "concise form" for writing uncertainties. I.e. $1.672\, 621\, 923\, 69(51)\times 10^{-27}$ kg instead of $(1.672\, 621\, 923\, 69\pm0.000\, 000\, 000\, 51)\times 10^{-27}$ kg.
- 1,538
4
votes
Accepted
How large does $N$ need to be for statistical mechanics to be a good approximation?
I think it's unfair to ask for an exact value for $N$ to justify all statistical mechanics. There are very many different problems and applications of stat mech, and some of them might have ...
- 2,104
4
votes
If a measurement has 5% error, can we say it has 95% accuracy?
You need to define what "error" means; typically it is an estimate of the standard deviation based on a series of measurements. If you take a series of measurements, you can estimate the ...
- 6,921
4
votes
Accepted
Determining the appropriate number of significant figures to report in least squares analyses
The most authoritative source for how to estimate and report measurement uncertainty is the BIPM’s publication: Evaluation of measurement data — Guide to the expression of uncertainty in measurement. ...
- 69.2k
4
votes
Why should a clock be "accurate"?
Most of the answers talk about being able to compare clocks, which is important, but not the fundamental issue. The point of having an accurate clock is to have an accurate and universal measurement ...
- 1,798
4
votes
Multiplication and significant figures
Actually, it becomes $4\times10^8$, not $400000000$. It seems pointless because you are thinking of exact numbers in your example, and if every number in there was an exact number it would be ...
- 7,745
3
votes
Can adding two numbers increase the significant figures?
Consider what the number of significant figures indicates:
The number written as $7$ has $1$ significant figure; the number is somewhere between $6.5$ and $7.5$
The number written as $8$ has $1$ ...
- 10.4k
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