Measurement Assignment

PRECISION, SIGNIFICANT AND DOUBTFUL DIGITS

Every measurement of a physical quantity contains a number and a unit. The precision of the measurement is limited by the measuring instrument used; a ruler without the small mm markings between the centimetres would not be as precise as a better quality ruler which showed the 10 mm markings between each centimetre. The better ruler would enable a measurement to be made such as 4.50 cm [right on the 5 mm (.5 cm) mark], while a cheaper ruler would only yield a measurement of 4.5 [or even 4.4 or 4.6] cm.

The last digit in any and every measurement is doubtful. The doubtful digit in the measurement is 0, the second decimal place, while the cheaper ruler’s doubtful digit is in the first decimal, the 5 (or 4 or 6 depending on the judgement of the measurer). The amount of doubt in all actual measurements is called experimental error or uncertainty.

The amount of precision in a measurement is reflected by the number of significant digits (or often the number of decimal places) in the number. Thus, 109.2 cm is a more precise measurement (4 significant digits) than 109 cm (only 3 significant digits). However, watch out for 0’s that function as place holders—a 100 kg bag of flour is probably measured roughly to approximately 100 kg (it might then be rounded to 100) and so the 100 kg measurement contains only 1 significant digit since the two 0’s are only placeholders. In scientific notation (1 × 10² kg) this becomes more obvious. If the measurement was made very precisely the measurement should read 100.0 or 100 kg (4 and 3 significant digits respectively). Only counts are exact since there is no measurement involved. Place the number of significant digits after the measurement and underline the doubtful digit.

a) 9.350 cm b) 10.039 g c) 6350 mg d) 0.006 km

e) 80 students f) 1009 mm g) 50 s h) 2.4 × 10³ kg

ACCURACY

Measurements are accurate if the measuring instrument is not damaged or worn out and if the experimenter (you!) uses good technique and judgement in making the measurement. Partners can work together to prevent a blunder from occurring. The accuracy of a measurement cannot be seen by the number of significant digits or decimal places, only by comparing the measurement to an accepted value (the “right answer”). If the line is measured by a careful experimenter to be 10.5 cm, a sloppy person’s measurement of 11.50 cm seems more precise but they just plain goofed (perhaps they forgot to subtract the “1” that they started from when they placed the line along their ruler).

If you had a choice of a measurement (or a measurement instrument) which had either excellent accuracy or excellent precision, which would you pick and why?
_______________________________________________________________________
_______________________________________________________________________.

AN EXAMPLE

Calculate the following to the proper number of significant figures (same number of significant digits as in your “worst” measurement) Calculate the area of the side of the block. Use 4 lines as follows:

1. Area of a rectangle
2. Area = l × w
3. = (_____ cm) × (____ cm) (Put your measurements)
4. = ____ cm²

Label the radius and diameter. Give the length of each in centimetres. Calculate the area of the circle to the proper number of significant figures. Use 4 lines for each calculation.

1. Area of the circle
2. Area = pr²
3. = 3.14 × (___ cm) × (___ cm)
4. = ____ cm²

Now find the areas of the following figures using the style showed in the above examples.

Volumes can be caluculated for rectangular blocks using V=l × w × h. The volume of a cylinder can be found using V = p r² × h. Find the volumes of the following.

		Measure the radius horizontally along the top of the cylinder.

METRIC CONVERSION

The metric system using a basic unit and a prefix to modify that basic unit.

The common prefixes are c, m and k (1/100, 1/1000, and1000). You can use the ‘magic’ of the number 10 to do quick metric conversions from one unit to another.

a) g to kg divide by _____ (move decimal 3 to the left)

b) kg to g multiply by _____ (move decimal___ to the ____)

c) m to cm multiply by _____ (move decimal___ to the ____)

d) cm to m divide by _____ (move decimal___ to the ____)

e) g to mg multiply by _____ (move decimal___ to the ____)

d) mg to g divide by _____ (move decimal___ to the ____)

The conversions above would also work for converting m to

km, or cg to g, or mm to m etc... It is the prefixes which count!

If the unit gets bigger, then the number gets smaller.

And if the unit gets smaller, then the number gets bigger.