17.41 Units in equations

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Most people have seen Einstein’s famous equation

E = mc2.                    (1)

But what does it mean? It means that if stuff with a mass m is completely destroyed, it changes into energy E, when c represents the speed of light in a vacuum. When I say, “completely destroyed”, I don’t broken or burnt (burning simply changes stuff into something else – post 16.33) – I mean that it ceases to exist. We’re not familiar with this happening in our everyday lives but the appendix gives an example of when this can happen.

Now suppose someone claimed that Einstein had made a mistake and the equation relating energy to mass should really be

E = mc.                    (2)

Would you believe it? You might think that it would be much too difficult to check this claim for yourself. But it isn’t!

We measure energy in joules (abbreviation J, post 16.21) which is given by a force, measured in newtons (abbreviation N), multiplied by a distance, measured metres (abbreviation m) – see post 16.20. (Don’t confuse the standard abbreviation for a metre, m, with the symbol I’ve chosen to use to represent mass, m; for more details see post 16.13). Force is the rate of change of momentum which is mass (measured in kg), multiplied by speed (metres per second, m.s-1) – see post 16.13. So force, the rate of change of momentum is measured in (kg.m.s-1)/s = kg.m.s-2. In other words 1 N = 1 kg.m.s-2. The beginning of this paragraph tells us that 1 J = 1 N.m = (1 kg.m.s-2).m = 1 kg.m2.s-2.

Now let’s look at equation 2. We measure mc in the units of mass (kg) multiplied by the units of speed (m.s-1), that is in kg.m.s-1. So equation 2 says that something measured in kg.m.s-1 is equal to something measured in kg.m2.s-2. This can’t be true. It’s rather like saying the distance between Paris and Montpellier is about 700 kg. Or like saying if I have 3 oranges and 2 apples and then eat one orange and one apple, I’m left with 2 bananas and 1 apricot. These examples don’t make sense! Neither does equation 2, for the same reason.

In Einstein’s equation (equation 1) mc2 is measured in kg.(m.s-1)2 = kg.m2.s-2 which is the same as a joule, the unit of energy. So this equation is believable.

Be careful! In the equation

E = 0.999mc2                     (3)

both sides are measured in the same units. But that doesn’t mean it is correct. Checking the units doesn’t enable us to detect the presence of any numbers like 0.999 or 1 or π (post 17.11) in an equation. Remember that trigonometric functions like sine (sin), cosine (cos) and tangent (tan) are also numbers (post 16.50), so we can’t use this method to tell if a trigonometric function is missing from an equation.

In post 17.14, we saw that there are other measurement systems as well as SI (in which mass is measured in kg, distance in metres and time is seconds (post 16.12). The same arguments should apply to these other measurement systems. So we could generalise the arguments used in this post by using M to denote the units used to measure mass, L to denote the units used to measure length an T the units used to measure time. Then the units of energy would be M.L-2.T-2, for the reasons given above. Sometimes M, L and T are called the dimensions used to measure things.

If you’re studying science, it’s worth checking the units of any new equations you meet. Textbooks and teachers (and even blog writers) sometimes make mistakes!

Related posts

17.39 Translational and rotational motion
16.28 Significant differences
16.25 Electrical charge
16.24 Accuracy and precision
16.12 Measuring movement
16.7 Writing numbers

 

Appendix

When some unstable atomic nuclei decay (post 16.6) they produce a particle called a positron. An example is the fluorine isotope F19 (post 16.27). A positron is identical to an electron but has a positive charge (posts 16.27 and 16.25). But a positron can only move a short distance before it meets an electron; they then destroy each other producing energy in the form of gamma radiation. The positron and electron are examples of anti-particles.

Is this just obscure physics? No. Many hospitals have PET (positron emission tomography) scanners. These can be used, for example, to detect regions with a high glucose metabolism by injecting the patient with glucose in which the -O-H group in the glucose molecule (post 16.30) is replaced by F19. The regions in the body that need a lot of glucose will then emit gamma rays that can be used to form an image. So matter is being destroyed to create energy in many big hospitals every day.

 

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