Fundamental and Derived Quantities
Physical quantities are not generally independent of one another. Many quantities can be expressed in terms of more fundamental quantities. The first three fundamental quantities we will deal with are those of mass, length and time. Many derived quantities can be expressed in terms of these three. For example, the derived quantity speed can be expressed as length/time.
Note that there are seven fundamental quantities in all. The other four are: current, thermodynamic temperature, amount of substance and luminous intensity. We will deal with these as we need them.
The expression of a derived quantity in terms of fundamental quantities is called the dimension of the derived quantity. The symbol M is used to denote the dimension of mass, as is L for length and T for time. So, for example, to determine the dimensions of the derived quantity speed, we would look at the formula for speed, namely: speed = distance/time. The dimensions of speed are then: [speed] = [distance] / [time] = L / T , where the square brackets are used to indicate that we are calculating the dimensions of whatever is inside the brackets. Finally, we use our knowledge of indices to simplify this expression. [speed] = LT-1
Since the precision of all measuring instruments is limited, the number of digits that can be assumed as known for any measurement is also limited. When making a measurement, read the instrument to its smallest scale division. Estimate within a part of a division. The figures you write down for the measurement are called significant figures.
In Physics, if you write 3.0, you are stating that you were able to estimate the first decimal place of the quantity and you are implying an error of 0.05 units. If you just write 3, you are stating that you were unable to determine the first decimal place and you are implying an error of 0.5 units. It is very important that you do not overstate the precision of a measurement or of a calculated quantity. A calculated quantity cannot have more significant figures than the measurements or supplied data used in the calculation. So, for example, if the length, breadth & height of a rectangular prism is each known to 2 significant figures, the volume calculated from these figures cannot have more than 2 significant figures. Let’s say the volume = 3.7cm x 2.9cm x 5.1cm = 54.723 cm3. You would state the volume as 55cm3 (2 significant figures only). Note that we have rounded the volume up to the nearest whole number in this case.
Zeros between the decimal point and the first non-zero digit are not significant. eg 0.00035 has 2 significant figures.
Zeros that round off a large number are not significant. eg 35,000 has 2 significant figures.
Zeros at the end of a string of decimals are significant. eg 0.5500 has 4 significant figures. The last 2 digits are meaningful here. The measurement is 0.5500 not 0.5501 or 0.5499.
Zeros in between non-zero digits are significant. eg 0.7001 has 4 significant figures. The first zero is not significant but the next two are.
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