13 Equations and symbols
The word ‘equation’ is used for an expression containing an equals sign. The quantities under consideration may be described in words, for example
in which case the equation is known as a ‘word equation’, or the quantities may be represented by symbols, for example
but the important thing to remember is that what is written on the left-hand side of the ‘=’ sign must always be equal to what is written on the right-hand side. Thus, as explained in Section 14.5, you should never use ‘=’ as a shorthand for anything other than ‘equals’.
You know (from Section 6.1) that, for squares and rectangles, the area is found by multiplying the length by the width and that, for rectangular block-like structures, the volume is found by multiplying its length by its width by its height.
Write word equations (i.e. expressions involving an equals sign like that given above for average speed) for the area of a rectangle and the volume of a rectangular block.
Area = length times width,
i.e. area = length width (or equivalently, area = width length).
Volume = length times width times height,
i.e. volume = length width height
Writing out all of these words is rather tiresome. However, equations can be expressed far more compactly if each word is replaced with a single letter, so the equation for the volume of a rectangular block might become:
where V represents volume, l represents length, w represents width and h represents height.
Rewrite the word equation , using letters instead of words.
The choice of letters was left up to you, so there is no single right answer. If you chose the symbol d to represent density, m to represent mass and v to represent volume, your equation will be d = . (Note that the symbol representing a quantity is usually written in italics. This is particularly important where the symbol is also used as an abbreviation for a unit. For example, m is used here to represent mass but m is the abbreviation for metre.)
It is reasonable to use the first letter of each quantity (e.g. m for mass). It makes the choice of letter more memorable and this is a perfectly acceptable answer. However, d for density might be confused with d for distance and v for volume might be confused with v for velocity. So, scientists try to reserve one letter for each commonly used quantity. Unfortunately, there aren’t enough letters in the alphabet, so it is conventional to use the Greek letter ρ (pronounced ‘rho’) to represent density and a capital V to represent volume, so the equation for density becomes:
Similarly, speed is conventionally represented by v (for ‘velocity’), so using d for distance travelled and t for time taken, the equation for speed becomes:
There is one more convention that you need to know about:
When using symbols instead of words or numbers, it is conventional to omit the multiplication sign, ‘’.
So, the equation for the volume of a rectangular block becomes:
where V represents volume, l represents length, w represents width and h represents height.
Equations 13.1, 13.2 and 13.3 will enable you to calculate density, average speed and the volume of a rectangular block.
Question 13.1
- a.An ice cube can be considered to be a rectangular block of ice with a length of 3 cm, a width of 2 cm and a height of 15 mm. What is the volume of such an ice cube?
where V is volume, l is length, w is width and h is height. In this case:
l = 3 cm
w = 2 cm
h = 15 mm = 1.5 cm
Substituting these values into the equation gives:
So, the volume of the ice cube is 9 cm3.
Alternatively, you could start by converting the length, width and height to metres:
l = 3 cm = 3 10-2 m
w = 2 cm = 2 10-2 m
h = 15 mm = 15 10-3 m = 1.5 10-2 m
Substituting these values into the equation gives:
So, the volume of the ice cube is 9 10-6 m3.
Since 1 cm = 1 10-2 m, 1 cm3 = (1 10-2 m)3 = 1 10-6 m3 so the answers obtained by the two methods are equivalent. An answer of 9 000 mm3 is also acceptable.
- b.The largest recorded iceberg in the Northern Hemisphere was approximately rectangular in shape with a length of 13 km, a width of 6 km and an average height of 125 m. What is the volume of the iceberg?
where V is volume, l is length, w is width and h is height. In this case:
l = 13 km
w = 6 km
h = 125 m = 0.125 km
Substituting these values into the equation gives:
So, the volume of the iceberg is 9.75 km3.
Alternatively, you could start by converting the length, width and height to metres:
l = 13 km = 13 103 m = 1.3 104 m
w = 6 km = 6 103 m
h = 125 m
Substituting these values into the equation gives:
So, the volume of the iceberg is 9.75 109 m3.
Since 1 km = 1 103 m, 1 km3 = (1 103 m)3 = 1 109 m3 so the answers obtained by the two methods are equivalent.
Now look at one final equation, which gives the volume, V, of a sphere of radius r:
where is a constant ( is the Greek letter pi, pronounced ‘pie’). The constant has a value of 3.141 592 654 (to 9 decimal places) but you don’t need to remember this as it is stored in your calculator – look for the button now! You can use Equation 13.4 to find the volume of any sphere anywhere in the Universe.
Equation 13.4 is more complicated than any of the other equations we have considered, so it needs a closer look. First, the multiplication signs have been omitted, i.e. the equation could be written as:
Also, note that the radius term, r, is cubed: the powers notation, already introduced for numbers and units, can be used for symbols too.
However, note that only the r is cubed, not the or the , so when you substitute values into Equation 13.4, you need to take care to calculate:
Raindrops are approximately spherical and they have a diameter of about 2 mm, i.e. a radius of about 1 mm. Using Equation 13.4, find the volume, in metre3, of a typical raindrop:
so
This is about 4 10-9 m3. Check that you can obtain this value for yourself, taking special care to cube both 1 10-3 and its units (m).
The Earth can be thought of as a sphere with a radius of 6.4 106 m. Use Equation 13.4 to find a value for the volume of the Earth.
Using Equation 13.4:
So, the Earth has a volume of about 1.1 1021 m3.
Question 13.2
Using Equation 13.4, what is the volume of a hailstone with a diameter of 1 cm? Write out your calculation in the style recommended in Section 14.5, showing each step of the calculation, and give your answer to one significant figure.
The diameter of the hailstone is 1 cm, so its radius is 0.5 cm.
So, the volume of the hailstone is 0.5 cm3 to one significant figure.
Alternatively, you might have started by converting the radius to a value in metres:
Substituting this value into Equation 13.4 gives:
The answers obtained by the two methods are equivalent. It is also reasonable to give a value for the volume of the hailstone in mm3 (500 mm3 to one significant figure).