# Problem-Solving - Significant Figures

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### Dimensions of Physical Quantities

When dealing with calculations of physical quantities, we need to be careful to not add "apples and oranges". The following table gives the dimensions for various physical quantities with L standing for a dimension of length, m for mass, and t for time. We will come across all of these in this course of study.

Quantity Symbol Dimension
Area A L2
Volume V L3
Speed v L/T
Acceleration a L/T2
Force F ML/T2
Density r M/L3
Energy E ML2/T2
Power P ML2/T3

In order to do any mathematical operation (add, divide, multiply, or subtract) with any of these physical quantities, we must be sure that all the physical quantities involved in the operation are of the same type. For example, it is wrong to do the operation

A + B = C

if A is a velocity and B is a force. The dimensions for C would be undefined. Many great ideas in physics were begun on the basis of making calculations that were dimensionally correct. For example, physicists had noted that two of the fundamental constants in physics that determine the strength of electric and magnetic interactions, when multiplied gave units which were the inverse square of a velocity. The precise number was exactly equal to the inverse square of the speed of light in fact. This was the first hint of a deep connection between what appeared to be very different physical phenomena: electricity, magnetism, and optics.

Dimensional analysis is also one of the methods for determining whether some error has occurred in the midst of a calculation since units must be consistent at all points.

### Significant Figures and Scientific Notation

All physical measurements are limited by intrinsic uncertainties in the method by which they are made. Even in the ideal exact measurement is not possible since ultimately quantum mechanics imposes fundamental limits on what we can quantitatively know about a system. Since the measurement technique determines the uncertainty in the measurement, quantitative sciences ONLY deal with measurements which are accompanied by the associated uncertainty. Hence it is never appropriate to state the result of a laboratory measurement as, say 3 centimeters. We are required to state the uncertainty as part of the measurement, e.g. 3 ± 0.2 cm. The 0.2 cm indicates that we can have made an error of as much 0.2 cm in the measurement of 3 cm so someone else making the measurement with a similar or more accurate instrument might get a value as low as 2.8 cm (but not likely a lower value) or a number as high as 3.2 cm (but not likely a higher value).

In making calculations, we need to keep in mind that the significant figures are determined by the last digit (other than the zeroes used to locate the decimal point) which is believed reliable. Hence, if our measurement of a speed is accurate to within 5 mm/s then we would state a measurement of speed in meters per second with 2 decimal places after the decimal point as in 4.52 m/s. A third decimal place after the decimal point would not be reliable as it could be off by as much as ± 5. The number of significant digits is determined by the number of digits which are reliable, therefore 4.52 has 3 significant figures. A number such as 121.83 has five significant figures and 0.00012 has two significant figures since the zeroes before the "12" only locate the decimal point.

As a matter of convenience for writing and a means of emphasizing the significant figures of a measurement, we use scientific notation for numbers. This notation states numbers as a product of a number and a power of 10. For example, the number 0.00012 in scientific notation would be stated as

1.2 × 10-4

We can apply a combination of significant figures, scientific notation and unit conversions to a typical problem.

Example: The SI unit of volume is the liter, defined as the volume that fits in a cube 10 cm along each dimension. What is the density of water (in g/cm3) if 2200 pounds occupies a volume of 1000 liters?

Solution:
Use the unit conversion tools as needed.

Problems:

1. There are 640 acres in a square mile. How many square cm are there in one acre?

2. Calculate the following rounded off to the correct number of significant figures. Express your result in scientific notation
1. (1.14)(4.998 × 106)
2. 3p/(8.96 × 106)
3. 2.78 × 10-8 - 5.698× 10-9
4. 42.1683 + 0.332

3. Complete the following calculations with the appropriate scientific units and notation. (You may need to refer back to the section 2 on units.)
1. power of a nuclear reactor = (103kg) (4.23 × 106 m2/s3)
2. lifetime of a subatomic particle (the muon) = (8.36 cm)/(3.8 × 104 m/s)
3. mass of the earth = (1.96 × 1030 J)/ (572.5 m/s)2