how to use units in calculations

8/10/2019 How to Use Units in Calculations

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How to use units in calculationsHere you can learn about how units should be used in calculations and with graphs. You will also seehow thinking about units can be helpful in spotting when you have got the physics wrong.

Physical quantity equals numerical value times unitWhat is the magnitude of the acceleration due to gravity? 10 or 9.81? Well, neither, actually! The

numerical value varies from place to place on the Earth's surface, of course. But even allowing forthis, you have to be very careful to include the units when writing down physical quantities. So, you

should write:

2sm81.9 g

Notice now that the symbol gstands for a number with its associated unit. You could use differentunits. For instance:

2scm981

g

A different value, a different unit, but the same g.

The number and the unit are effectively multiplied together. Thus the acceleration due to gravity is

9.81 times the unit m s-2.

Equations, numbers and unit s

Now think about how this applies to equations. When an object falls in a gravitational field it does sobecause of the force of gravity upon it. This is called its weight, which you can calculate from the

equation:

mgW

where mis the mass of the object. If you were asked to calculate the magnitude of the weight of an

object of mass 5.00 kg, you might write:

N1.4981.900.5 mgW

As written, although the answer is correct and has the correct unit, the mathematical 'sentence' as a

whole is incorrect. The sentence begins with the symbol 'W ', standing for the magnitude of the weight

- a numerical value multiplied by its associated unit. The next part of the sentence, mg, also includes

numerical values and units. However, the arithmetic part of the sentence, 5 9.81, contains only

numerical values. The units are missing. So, rewriting the mathematical sentence with the correctgrammar, you should have:

N1.49sm81.9kg00.5 2 mgW

Only if you include the units at all stages in a calculation will your mathematical grammar be correct.

The final equals sign now tells us two things: that 5.00 9.81 is numerically equal to 49.1 (to the

accuracy of the data) and that the unit kg m s-2is equivalent to the newton, or:

N1smkg1sm1kg1 22

Graphs and units


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Axes, gradients and areasIn chapter 8 of theAdvancing Physics ASstudent's book you can read about distance-time and

speed-time graphs. These sorts of graphs are used here as examples of how useful thinking aboutunits can be in graphical work.

Look at the speed-time graph below. It might be of a short (if unrealistic) bicycle ride. First, look at the

way the axes are labelled. Only numbers, without units, are written on the axes. The quantities are

written as 'speed / m s-1' and 'time / s'. The reason for this particular way of labelling the axes is that

the word 'speed' stands for a number and its associated unit in the same way that the symbol 'g'

does. If the quantity 'speed' is divided by the unit 'm s-1', you are left only with a number, which is

what appears on the axes.

Now, when interpreting graphs you often need to calculate gradients and areas - these may representimportant physical quantities. Their units can give an important clue as to what they represent.

0 5 10 15 20

0

2

4

6

8

10

12

time / s

pedalling my bike

area = height width

The graph shows someone pedalling from rest to a speed of 8 m s -1in 10 s and then pedalling at a

steady speed of 8 m s-1for the next 10 s. The first section of the graph shows the speed changing.Work out the gradient: pick two points on the straight line section of the graph and then divide 'up' by

'across'. The unit of the gradient must be that of 'up', m s-1, divided by that of 'across', s. Thus, theunit of the gradient is

2

2

1

sms

m

s

sm

and this is the unit of acceleration. Well, this is not a surprise: after all, you can see that the first part

of the graph shows the bicycle speeding up. The point, though, is that even if you did not know the

significance of the gradient, the units tell you something about its physical meaning.

What about the area? Part of this is shaded dark blue on the graph, the rectangular part whose area

is labelled height width. This is not the whole area, of course, but the unit of any part of the area

must be the same as the unit of the whole area. Therefore, the unit of the area must be that of the

width of the rectangle, in s, multiplied by that of the height, in m s-1. Therefore, the unit of the area is

mss

mssm 1

The area has the same unit as distance! So the area under a speedtime graph gives the distance

travelled.


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Units and equations

The simple pendulumSuppose that you know there is a formula for the period of a pendulum, and that it involves the length

l of the pendulum in metres, and the acceleration gdue to gravity in metres per second squared.

Unfortunately, you can't remember the details of the formula; all you can remember is that the formula

for the period, T, is:

glT andofncombinatiosome2

With two quantities, l and g, you can add, subtract, multiply or divide. As l and ghave different units,

addition and subtraction are impossible you cannot add metres to metres per second squared any

more than you can add apples to oranges! So try multiplying the units of l and g:

222 smsmm

This does not look much like a time. Try division; the units of (l /g) are:

2

22s

s

1

sm

m

This looks more like it. If you take the square root of this, you will have a time:

ss2

The combination of l and gthat gives the period of the pendulum must be

gl

so that the formula is:

g

lT 2

Notice that thinking about units tells you nothing about the constant, 2, as this has no units. You

simply have to remember that this dimensionlessconstant appears in the formula. Bear in mind,

however, that not all constants are dimensionless. For instance, the extension of a spring when a

force is applied to it is given by the Hooke's law equation:

kxF

The spring constant, k, has units of N m-1, as you should be able to see by thinking about the units of

Fand x.

Fundamental unitsSo far, all the examples have used units of mass, length and time only. Is it really all this easy? No, it

isn't. How many fundamental units are there that you need to know? In the SI system of units, thereare seven fundamental units from which all the others are derived. These are shown in the table

below.

Quantity Unit Symbol

Length metre m


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Quantity Unit Symbol

Mass kilogram kg

Time second s

Electric current ampere A

Temperature kelvin K

Luminosity candela cd

Amount of substance mole mol

Other quantities can all be expressed in these fundamental units. For example:

The unit of force, that is of mass acceleration, is the newton N = kg m s-2

The unit of energy, that is of force distance, is the joule J = N m

The unit of power, that is energy per second, is the watt W = J s-1

The unit of charge, that is, of current time, is the coulomb C = A s

The unit of potential difference, that is of energy per coulomb, is the volt V = J C-1

The unit of conductance, that is of current per volt, is the Siemens S = A V-1

The unit of resistance, that is volts per ampere, is the ohm = V A-1

Checking equationsYou can use units to check that equations make physical sense. For example the power dissipated in

a circuit is given by:

P =IV

Power, of course, is measured in watts and you know that:

1sJ1W1

Now check that the units of the expressionIVare also J s-1. The unit of current is the ampere A, and

the unit of voltage V = J C-1, so you can write the unit of the expressionIVas A J C-1. Notice that

since the coulomb C = A s, the unit ofIVis also A J [A s]-1, which reduces to J s-1, as it must.

The units on both sides of an equation must match. If they do not, the equation is wrong.

In conclusion

You have seen that the units of physical quantities are important and that you should make everyeffort to include them at all stages in a calculation. You have also seen how graphs should be labelled

and how thinking about units can point the way to important physics that may be hidden in thegradients or areas of graphs. You have also seen that checking that the units agree on both sides of

an equation can very quickly tell you when things have gone wrong.

We hope that, from now on, whenever you see a graph you will think about the units of the gradient

and area and whether or not they are significant; we hope too that you will check the units ofequations at important stages in a derivation or calculation (especially your own!) and that you will


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make the effort to include the units at all stages in a calculation. Doing all this can make physics

clearer to you and to those who read your work.

how to use units in calculations

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