This experiment is motivated by a companion experiment with matchstick rockets. It would be nice to be able to accurately weigh a matchstick rocket, both before and after ignition. From that we might be able to estimate the weight of the gases burned during the flight. But matchstick rockets are pretty light. How do we make an accurate measurement of something so lightweight?
The only scale readily at hand is the Cuisinart Kitchen scale shown in the picture above. How precise and accurate is this scale? How do we use the scale? To find out, we need to experiment. It is often quite common that we need to experiment with our measuring equipment to learn how to use it, how to calibrate, what precision it has, and how accurately (or to what extent we can trust it).
The scale reads in ounces or grams, and toggles between the two by pressing the green button. The red button turns the unit on, and also execute a tare weight correction (link to tech note on tare weight). We also need something to weigh. I got about 25 dimes together to use as a standard weight.
According to the US Mint, (see: http://www.usmint.gov/about_the_mint/?action=coin_specifications ) the standard weight of a dime is 2.268 grams. So If we weigh one dime we should get that value. Weight 2 dimes, and we get double that value, and so on. Right? Let's find out.
If you look closely at the picture above you will see that there are 3 dimes on the scale, but the display reads 0 grams. This is after the tare button was pressed with 0 dimes. Huh?
OK so let us run this to ground. I got a 9 year old to help me make some measurements. We weighed a series of dimes, and did a tare correction before each measurement. Here are some results:
#of Dimes Display (grams) Weight per dime (grams)
1 0 -
2 0 -
3 0 -
4 5 1.25
5 10 2.0
10 20 2.0
15 35 2.33
20 45 2.25
25 55 2.2
OK, what have we learned? First, notice that the scale only reads out in increments of 5 grams. So weighing a number of dimes, and dividing by the quantity is one way to get a more precise weight per dime.
Next notice that something funny is going on at the low end. Three dimes should weigh 6.804 grams altogether, yet the scale still reads zero. Is the scale sticky? Would jiggling it a bit help?
OK, now let's look at the third column. This is just the second column divided by the first. The accuracy of the measurement seems to get better as the number of dimes increases, yet the result for 20 dimes is closer to the truth that the result for 25 dimes. Why? I don't know the answer, but here are some hypotheses:
-there may be random noises at play. We could repeat the weighing of 25 dimes a number of times and compute the variance of each measurement. Let's do that. After 10 trails of weighing 25 dimes, we get a reading of 55 grams every time. OK, so no random noise in the measurement?
-there may be a systematic error. For example, let say that we made a mistake counting out the number of dimes. Instead of 25 dimes, we accidentally weighed only 24 dimes. That would have given us a weight per dime of 2.29 grams. That is a lot closer to the true value. So is that what happened? Not so fast. Just because your hypothesis gets you closer to the true value, doesn't mean it is correct. What if you did not know the "true" weight? Anyway, it is pretty easy to repeat the experiment and double and triple check the count. Let's do that. Nope, 25 dimes weighs 55 grams even after double and triple checking the dime count.
- quantization of the display value to the nearest 5 grams, is a source of error. An estimate of the error due to quantization is 5 grams divided by the square root of 12, or 1.58 grams. (this is based on some fancy statistical analysis which assumes a uniform distribution of weight). We would expect to get no better than 1.58 grams accuracy with any single weighing event. But this error gets divided by the number of dimes, so divided by 25, gives an error of 0.06 grams. This gets us pretty close to the true value (but we don't know which way). So we could say, that after weighing 25 dimes, the weight of one dime is 2.2 grams with an uncertainty of 0.06 grams.
-the scale may have non-linear response. If this is so, then our assumption of weighing multiple dimes, and then dividing by the number, will not be valid. It appears that this may be so at the low end of the scale.
Anyway, all of this is moot, since our scale does not appear to be trustworthy with anything less that 10 or more dimes (or 20 grams). An a matchstick rocket will weigh much less than a dime. So we are stuck.
The only scale readily at hand is the Cuisinart Kitchen scale shown in the picture above. How precise and accurate is this scale? How do we use the scale? To find out, we need to experiment. It is often quite common that we need to experiment with our measuring equipment to learn how to use it, how to calibrate, what precision it has, and how accurately (or to what extent we can trust it).
The scale reads in ounces or grams, and toggles between the two by pressing the green button. The red button turns the unit on, and also execute a tare weight correction (link to tech note on tare weight). We also need something to weigh. I got about 25 dimes together to use as a standard weight.
According to the US Mint, (see: http://www.usmint.gov/about_the_mint/?action=coin_specifications ) the standard weight of a dime is 2.268 grams. So If we weigh one dime we should get that value. Weight 2 dimes, and we get double that value, and so on. Right? Let's find out.
If you look closely at the picture above you will see that there are 3 dimes on the scale, but the display reads 0 grams. This is after the tare button was pressed with 0 dimes. Huh?
OK so let us run this to ground. I got a 9 year old to help me make some measurements. We weighed a series of dimes, and did a tare correction before each measurement. Here are some results:
#of Dimes Display (grams) Weight per dime (grams)
1 0 -
2 0 -
3 0 -
4 5 1.25
5 10 2.0
10 20 2.0
15 35 2.33
20 45 2.25
25 55 2.2
OK, what have we learned? First, notice that the scale only reads out in increments of 5 grams. So weighing a number of dimes, and dividing by the quantity is one way to get a more precise weight per dime.
Next notice that something funny is going on at the low end. Three dimes should weigh 6.804 grams altogether, yet the scale still reads zero. Is the scale sticky? Would jiggling it a bit help?
OK, now let's look at the third column. This is just the second column divided by the first. The accuracy of the measurement seems to get better as the number of dimes increases, yet the result for 20 dimes is closer to the truth that the result for 25 dimes. Why? I don't know the answer, but here are some hypotheses:
-there may be random noises at play. We could repeat the weighing of 25 dimes a number of times and compute the variance of each measurement. Let's do that. After 10 trails of weighing 25 dimes, we get a reading of 55 grams every time. OK, so no random noise in the measurement?
-there may be a systematic error. For example, let say that we made a mistake counting out the number of dimes. Instead of 25 dimes, we accidentally weighed only 24 dimes. That would have given us a weight per dime of 2.29 grams. That is a lot closer to the true value. So is that what happened? Not so fast. Just because your hypothesis gets you closer to the true value, doesn't mean it is correct. What if you did not know the "true" weight? Anyway, it is pretty easy to repeat the experiment and double and triple check the count. Let's do that. Nope, 25 dimes weighs 55 grams even after double and triple checking the dime count.
- quantization of the display value to the nearest 5 grams, is a source of error. An estimate of the error due to quantization is 5 grams divided by the square root of 12, or 1.58 grams. (this is based on some fancy statistical analysis which assumes a uniform distribution of weight). We would expect to get no better than 1.58 grams accuracy with any single weighing event. But this error gets divided by the number of dimes, so divided by 25, gives an error of 0.06 grams. This gets us pretty close to the true value (but we don't know which way). So we could say, that after weighing 25 dimes, the weight of one dime is 2.2 grams with an uncertainty of 0.06 grams.
-the scale may have non-linear response. If this is so, then our assumption of weighing multiple dimes, and then dividing by the number, will not be valid. It appears that this may be so at the low end of the scale.
Anyway, all of this is moot, since our scale does not appear to be trustworthy with anything less that 10 or more dimes (or 20 grams). An a matchstick rocket will weigh much less than a dime. So we are stuck.