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Wize University Physics Textbook (Master) > Foundations / Introduction / Measurement

Conversion of Units

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Unit Conversion: multiplying by "one"

There are many different unit systems aside from the SI units. Below is a table of common units and their SI unit equivalent. Most of them you have probably heard of before.

When we are converting between units, we can multiply by a conversion factor, which can be thought of as cleverly multiplying by "one".

Examples

1. We have defined that 1m=3.28ft1m = 3.28ft , then 1m3.28ft\frac{1m}{3.28ft} and 3.28ft1m\frac{3.28ft}{1m} must both be equal to 1. If I am 6ft6ft tall, how tall am I in metres?





Notice how the unit of ftftcancel with the denominator, and we are left with our answer in mm . And when we multiply by one, we don't change the actual quantity of 6ft6ft, we only change the units. This is how we do unit conversions.

Wize Tip
Try to always write out the conversion units, and cancel out the units as you go. It is a super common mistake to accidentally flip the conversion units.

2. My phone has a 5.9in screen. How big is that in millimetres? (1m = 39.37in)





Here, I multiply by two conversion factors. One to get from inches to metres, and another to get from metres to millimetres.


Exam Tip
Think about if your answer makes physical sense! If your object is 5.2mm big, and you get a final number of 52000km... you probably made a mistake in your conversion.

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Here is some practise for unit conversion!

1. How many seconds are in a year?


1 year =1 year⋅365 days1 year⋅24 hours1 day⋅60 min⁡1 hour⋅60 sec⁡1 min⁡=31,536,000 sec⁡1\ year\ =1\ year\cdot\frac{365\ days}{1\ year}\cdot\frac{24\ hours}{1\ day}\cdot\frac{60\ \min}{1\ hour}\cdot\frac{60\ \sec}{1\ \min}=31,536,000\ \sec



2. The speedometer in your car is in km/h. You go on a road trip to the US, where speed limits are posted in mph (miles per hour). You're on a highway with a 70 mph speed limit. If you are driving 110 km/h, are you speeding? (1 mile is 1.61 km.)

We need to convert our speed to mph.
110 kmh=110 kmh⋅1 mile1.61 km=68.2 mph110\ \frac{km}{h}=110\ \frac{km}{h}\cdot\frac{1\ mile}{1.61\ km}=68.2\ mph
This is less than the posted speed limit. We are not going to get a ticket!



3. I have a giant balloon. And it can hold up to 54100 human brains. Now a human brain is actually a measure of volume (1 human brain = 0.54ft30.54ft^3). There are 3.28ft=1m3.28 ft = 1m , so what is the volume of the giant balloon in m3m^3?

Convert to cubed feet first, and then convert to metres.
54100 human brains⋅0.54ft31 human brain⋅(1m)3(3.28ft)3=827.88m354100 \textrm{ human brains} \cdot \frac{0.54 ft^3}{ 1 \textrm{ human brain}} \cdot \frac{ (1m)^3}{(3.28 ft)^3} = 827.88 m^3
Here, notice how we scale our length (feet and metres) into volumes (feet3 and metres3).
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If a car is travelling at 72 km/h, what is its speed in m/s?

72kmhrx?  =ms\frac{72km}{hr}x?\ \ =\frac{m}{s}

We need to change km->m and hr->s

It doesn't matter which one we change first, let's do km->m

72kmhrx 1000m1km\frac{72km}{hr}x\ \frac{1000m}{1km}

Now let's change h->s

72kmhx 1000mkmx1h3600s\frac{72km}{h}x\ \frac{1000m}{km}x\frac{1h}{3600s}

=20m/s
How many times does your heart beat in a year?