Lab 1 Uncertainty and Error Propagation Introduction The ability to adequately r

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Lab 1 Uncertainty and Error Propagation Introduction The ability to adequately represent measurements is critical in science experiments. This lab is meant to teach the process of correctly taking measurements, procssing data, finding the range of error, and comparing them to a set value. Also, the propagation of error is introduced in determining the volume of various objects To begin with, we need to characterize the two types of experimental errors that occur, systematic errors and random errors. Large systematic errors can and must be eliminated; small systematic errors will always be present. For example, thermometer poor contact; cloudy or shady when measure radiation from Sun. Random errors are due to noise or imprecise definition. While in a careful experiment we can minimize the effect of random error, we can never completely remove it. Significant Figures The number of digits (or significant figures) used when reporting an experimental value that is either directly measured or determined from measurements, is a reflection of how confident we are in that value based on the measurement tools used. In other words, the number of significant figures represents the precision with which a measurement is performed based on the equipment used. To determine the number of significant figures in a measurement there are a few simple rules to follow: Nonzero digits are always significant; Final ending zeroes written to the right of the decimal are significant. Zeroes written to the right of the decimal point for the purpose of spacing the decimal point are not significant; Zeroes written between significant figures are significant. To determine the appropriate number of significant figures for the value of a quantity determined based on measured values, there are a few simple rules to follow. When multiplying and dividing, the number of significant figures in the final result is the same as the number of significant figures in the least precise of the factors being combined. When adding or subtracting, we need to round the result to the smallest number of decimal places of any term being combined Uncertainty and Error Propagation While in some types of experiments, the uncertainty in the value of a quantity is in fact solely based on the precis in different values each time they are performed. In circumstances where we want to report the value of that quantity or the value of any quantity determined using it while not overstating our confidence in the value, we should also report our uncertainty in that quantity ion of your measurement tools. However, in many experiments, there are measurements that will result To properly state the value of a quantity, we can repeat its measurement many times and average the values in the usual way. The mean (or average) value for a quantity is given by

Explanation / Answer

Significant figures can be a little trickc but once you practice it's rules regularly in your lab it will come second natured to you.

Let's say you use a standard meter scale to make a measurement of length (l) width (w) and height (h) of a cuboid.

The significant figures of this measurement will be 1mm , since that is the smallest unit measurable using a standard meter scale (the smallest division on the scale). This is called the least count of a measuring instrument.

1 mm = 0.1 cm

And suppose you get the measurements as:

l = 8.4 cm

w = 2.6 cm

h = 5.8 cm

So the volume is: V = l X w X h

V = 128.672 cm³ as you can see we are here claiming a degree of precession of 0.001 cm while our meter scale can be accurate upto only 0.1 cm.

So we are certain only about 128.7 cm³. (Rounding it iff to 0.1 cm precision)

Now suppose we had to do the same with a cylindrical object. Here we would use a Vernier caliper to find the diameter of the cylinder.

The Vernier calipers have a least count of 0.01 cm and we get our readings as

Radius (r) = 2.35 cm

Height (h) = 8.3 cm

So the volume computes to be :

V = pi X r² X h = 3.14*2.35² *8.3

V=143.927395 cm³

Here since we had multiple least counts the significant figures are decided by the least significant figure value.. ie of the meter scale 0.1 cm

So V = 143.9 cm³

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