Measuring Devices

Before I begin, let me discuss the tools we used to take the measurements. The vernier caliper is a measuring device that allows one to take the length of objects. It is specifically useful to take the diameters of round objects as it has provisions and clamps to both accurately measure inner and outer diameters. Some vernier callipers also allow depth to be measured. A vernier caliper comes in two parts, the main scale (MS) and the fractional scale (FS). The MS works just as any ruler would, with the 0 mark on the FS being the mark you base your reading off of. Like a ruler, you're measurement won't lie exactly on one number on the MS – this is where the FS comes in. The FS is used to increase the precision of the measurement. There are many lines on the FS, each corresponding to a different fractional unit (in our 1 mm vernier caliper, the FS came in gradations of 0.02 mm). You have to find the first line on the FS that matches with a line on the MS. That is your FS reading. To get your total measurement, you add your MS reading with your FS reading.

The micrometer caliper works the same way, except that it has a circular MS and FS and its FS has gradations of 0.01 mm as the MS has gradations of 0.5 mm. You put the object in between the appropriate section and turn the dial until you hear a click that signifies that the caliper is tightened. Once this happens, you read the mark on the MS, and then you check the number on the FS that "hits" the marking line. Adding these two values up, you get your main reading.

One thing of note is that you have to make sure your instruments do not have any zero error. This is the error caused by an uncalibrated caliper and can lead to systematic error. To check for zero error, you put your caliper all the way to its lowest measurement and see if the 0 on the MS aligns with the 0 on the FS. If it does, then you are good. If it does not, the take note of how much it differs and account for this in recording your measurements. Luckily for us, there was no zero error for either of our instruments.

Measuring Rice

Actually measuring rice was a bit tricky. For the vernier caliper, it took us a while before we managed to get the rice properly between the "teeth" of the vernier caliper. It was difficult to ensure that what we were trying to measure was the entire length of the rice grain. It proved even more so with the micrometer caliper as we had to place the rice in between a rather thick compression part. Nevertheless, after a few tries, we managed to get a hang of it and proceed with our measurements.

The following is a link to our measurements and the subsequent sub results to the questions:

https://docs.google.com/spreadsheets/d/1DsqfxJWet7PCCeKjzrh4j0L5fUz7opAenDcDDEFlfLk/edit?usp=sharing

*note: the measurements listed do not show the proper significant digits of 3 for all readings as a result of limitations on excel

1. What is the length of a rice grain?

The length of a rice grain varied in between a range of measurements from 6.33 to 8.00 mm from a total of 110 different measurements of different samples. The uncertainty of the measurements varied between the two instruments used, with ±0.01 being the uncertainty of the vernier caliper and ±0.005 being the uncertainty for the micrometer caliper. This answer was obtained through direct measurement of said samples. The validity of the measured values is known to be true as it was observed firsthand by the researchers.

2. What is the average length of a rice grain?

The average length of a rice grain is 7.14±0.86 mm. This answer was obtained through the formula 

AVERAGE = (SUM OF ALL MEASUREMENTS)/(TOTAL NUMBER OF MEASUREMENTS)

The uncertainty for this was taken as the maximum difference between a measured value and the mean of the set of data values.

It is known to be true because it follows a certain formula that utilises the measured data to produce its result.

3. What is the acceptable range for the length of a grain of rice?

The acceptable range for the length of a grain of rice, found through the formula Q=μ±σ, is  7.14±0.32 mm. The mean is taken as the average of the lengths of the grains of rice. The standard deviation, within which 67% of all measurements are found in, is taken through an equation utilising excel.

The precision can be commented on by comparing this to the known value of the length of a grain of this particular type of rice, l=7.09±0.36 mm taken from Hobson, Carter IMTC 2007 paper. As you can see, the values found are close to the known value, making the measurements accurate, the value for acceptable range being the most accurate as it encapsulates the known value of the actual measurement. It should also be noted that the questions are being asked in increasing orders of approximation which results in asking for an increasing level of precision.

As I mentioned in my previous post, in physics experimentation, there are a lot of measurements that we have to take. Measurement is the fundamental tool for gathering data so undoubtedly a lot of it is necessary as we test different hypotheses in the different studies within the field. When taking measurements, we need to note that what we are doing is finding quantitative descriptors about the objects we are looking at. Take a ball for example. We can describe it as red and bouncy, but that does not give us as much info as saying it has a diameter of 40.0 cm and a mass of 8.33 g in Physics.

What is a measurement?

A measurement comes in two parts, the magnitude and the unit. Take for example the 40.0 cm diameter of the ball. We have 40.0 as the magnitude. The magnitude is the number in a measurement. This tells us how much of a given quantity is present.Then we have cm as the unit. This tells us what aspect of an object we are measuring. Centimetres (cm) is a unit for measuring length. Other units are available for measuring length such as meters (m) and inches (in). Units are also present for other factors we wish to measure: there's seconds (s) for time, kilograms (kg) for mass, and many others.

Measurements are inherently uncertain. When measuring an object and seeing that it is 40.0 cm, how are we not sure that it is not actually 40.01 or 39.99 cm? Or how can we tell apart 40.0 cm from 40.0000000000000001 cm? We can with a very high precision measuring device, but for the most part, we can't. However, there are some conventions that we follow to properly show what we are certain and uncertain of in a measurement.

Given a 30 cm ruler with 1 cm gradations (a gradation is a line or a mark on the measuring device), we can measure an object surely until the smallest gradation. However, we know that sometimes, the object falls in between two lines, and we're not sure how to report the measurement. Say the object is between 5 and 6 cm. We can say that the object is 5.4 cm long, the .4 being the uncertain measurement. However, we cannot say that the object is 5.42367 cm long as we are unsure of even the first decimal number. The convention is noting measurements is that we are allowed one (1) uncertain digit past the smallest certain digit provided by the measuring device.

Error in Measurement

As we gather data though, we have to be aware that we are limited, both in ourselves as researchers and in the instruments that we use. Error is a major part in dealing with measurements. Slight mistakes to measurements taken, although seemingly small at first, can have great implications on experimentation as a whole. When scientists wanted to land a probe on a comet, they had to be exact with the measurements they took: time for the comet to travel around the sun, distance of the comet from Earth at given points in time, etc. 

There are two types of error that can be made when taking measurements. The first is random error. This is the error that comes from the circumstances in which the measurement was taken. This is mostly attributed to human error caused by the researcher taking the measurement. Sources of random error include how the researcher views the measurement along the gradations and how the researcher handles the measuring device, to name a few. The second type of error is systematic error. This is the error that comes from the experiment itself. This could include outside factors entering the set-up, an apparatus was slightly askew, or the measuring device was poorly calibrated.

Although error cannot be avoided, there are certain steps that can be taken to minimize their effects on the results of the experiment. Mistakes of random error cannot be avoided. However, since they are caused by random events and vary from one testing to another, a way to decrease the influence of random error is to increase the number of trials done. Mistakes of systematic error, on the other hand, can be avoided. By ensuring the experiment is in optimal conditions so that it is not affected by external factors, apparatuses calibrated properly, and experiment conducted properly, systematic error can decrease.

Precision and Accuracy

Another integral concept of measurements is differentiating between precision and accuracy. These descriptors are determined by measurements' distance away from each other and a known value. Measurements are precise if the values taken from them are close to one another. A measurement can be more precise if it has more decimal places. Measurements are accurate if they are close to the known value.

Levels of Approximation

Lastly, we cannot be infinitely precise with our measurements, and thus, we have to give way to certain levels of approximation.

The lowest level of approximation is orders of magnitude. This level of approximation utilizes powers of ten (10) to provide an approximation. Using this, we can say that an apple is 100 g or 10^3 g (using scientific notation). We cannot provide specific numbers as these are only general estimates that we conjecture from observation.

The first level of approximation is significant figures. Having the same apple as an example, we can say that the apple is 1.01 x 10^3 g.

Higher levels of approximation are referred to as best estimates.  A best estimate for a measurement Q is given in the form of Q=⟨Q⟩±ΔQ, where ⟨Q⟩ is the expectation value of the measurement and ΔQ is the uncertainty, where ±ΔQ is a range away from ⟨Q⟩ where the true value of the measurement lies.

There are several cases for best estimates on what determines both ⟨Q⟩ and ΔQ.

If the measurements being taken are regular or non-fluctuating, where repeated measurements produce the same value for the magnitude, then ΔQ is given as half the least gradation. In a 1 mm gradation ruler, the uncertainty ΔQ is ±0.5 mm.

If the measurements being taken are fluctuating and there are a limited number of them, the ⟨Q⟩ is given as the average of all taken measurements and ΔQ is the maximum difference between ⟨Q⟩ and any value within the set of measurements.

If the measurements being taken are fluctuating and there are a very large number of them,then Q=μ±σ. This is due to the idea that multiple measurements tend to follow a Gaussian normal distribution, with μ being the central value or average and σ being the standard deviation away from the central value.

With all of this knowledge of measurement in hand, I can be ready to face on the reset of the things I would learn in this class.

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