Physics · Class 9-10 · Chapter 1 · Physical quantities and measurement
Dimension.
In the last note you counted how many lengths, masses and times go into each derived quantity. Those counts have a name: dimensions. Once you have the name you get a new power with it. You can look at a formula and say it is wrong without substituting a single number.
1. The definition and the symbols
The dimensions of a physical quantity are the powers to which the base quantities appear in it. They are written inside square brackets. Length is [L], mass [M], time [T], electric current [I], temperature [θ]. Nearly everything at this level is done with just M, L and T.
Get one distinction straight at the start: dimensions are not units. 2 m, 200 cm and 7.8 feet have three different units and one dimension, [L]. The dimension tells you what kind of thing it is; the unit tells you against which standard it was measured.
Working them out is simple too. Write the defining formula, then put each quantity's dimension in place of the quantity. Velocity is displacement over time, so [v] = [L] ÷ [T] = [LT⁻¹]. Nothing to memorise; you can rebuild it every time.
2. The same table, now with a name
You have seen this grid once already. Last time it was only counting: how many times each base quantity is needed. Now those same numbers have a name, dimensions. There is nothing new to learn here; the familiar thing is just being called by its name.
gridtap a row
The table of dimensions
quantity
[M]
[L]
[T]
area
০
২
০
volume
০
৩
০
velocity
০
১
−১
acceleration
০
১
−২
force
১
১
−২
momentum
১
১
−১
work
১
২
−২
power
১
২
−৩
pressure
১
−১
−২
density
১
−৩
০
Area is length times breadth, so [L][L] = [L²].
Volume is length times breadth times height, so [L³].
Velocity is displacement over time, so [L] ÷ [T] = [LT⁻¹].
Acceleration is change of velocity over time, so [LT⁻¹] ÷ [T] = [LT⁻²].
Force is mass times acceleration, so [M][LT⁻²] = [MLT⁻²].
Momentum is mass times velocity, so [M][LT⁻¹] = [MLT⁻¹]. One power of T less than force.
Work is force times displacement, so [MLT⁻²][L] = [ML²T⁻²].
Power is work over time, so [ML²T⁻²] ÷ [T] = [ML²T⁻³].
Pressure is force over area, so [MLT⁻²] ÷ [L²] = [ML⁻¹T⁻²]. Notice that L goes 1 − 2 = −1.
Density is mass over volume, so [M] ÷ [L³] = [ML⁻³].
You do not need to memorise the table, because every row can be rebuilt from a defining formula in a couple of steps. Below, the dimensions of power are worked out in four, and it is worth noticing which direction the work goes.
step by stepwalk the steps
The dimensions of power, in four steps
Step 1. Start from the definition: power is work over time. So to get the dimensions of power you first need the dimensions of work.
Step 2. Work is force times displacement, so now you need force. Force is mass times acceleration, and acceleration is [LT⁻²]. So [F] = [M][LT⁻²] = [MLT⁻²].
Step 3. Now work: [W] = [MLT⁻²] × [L] = [ML²T⁻²]. The power of length goes up by one, to 2.
Step 4. Finally divide by time: [P] = [ML²T⁻²] ÷ [T] = [ML²T⁻³]. Those are the dimensions of power.
Notice that the work did not go forwards, it went backwards. Power needed work, work needed force, force needed acceleration. That is the method for any derived quantity: keep breaking it down until you land on base quantities.
You can also read it the other way: fix the three powers and see which quantity comes out. Change the powers of M, L and T below. Some combinations give something familiar, some give nothing at all, and that is an answer too.
the other way roundchange a power
Build a dimension
3. Same dimensions, same family
Two quantities with different names can share a dimension, and when they do it means they are the same kind of thing. Weight is a force, the tension in a wire is a force, friction is a force, so all three are [MLT⁻²]. In the same way work, energy and heat are all [ML²T⁻²], which is the clue that all three are forms of one thing.
sort themchip, then bin
Same dimension, same box
Pick up a quantity, then drop it in the box for its dimension.
Right. Next one.
Not there. Ask what the thing actually is: a force, an energy, or a speed?
Different names, same dimension, same family. Weight is a force and so is tension, which is why both are [MLT⁻²]. But be careful: sharing a dimension does not make two quantities the same thing. Work and heat share one, and work is not heat.
check yourselfpick one answer
Which two share a dimension
Which two of these have the same dimensions?
Right. Work, energy and heat are all [ML²T⁻²], because all three are forms of the same thing.
Pressure is force over area, so its power of L is two lower than force's. Force is [MLT⁻²], pressure is [ML⁻¹T⁻²].
This is the commonest confusion of all. Mass is a base quantity, [M]. Weight is a force, [MLT⁻²].
Acceleration has one more power of time than velocity. Velocity is [LT⁻¹], acceleration is [LT⁻²].
4. The ones with no dimensions at all
A number has no dimensions. 2, π, ½, any constant at all, are [M⁰L⁰T⁰]. That is why the ½ in s = ut + ½at² drops out of a dimensional check entirely. For the same reason some quantities have no dimensions either, and there is a pattern to them: nearly all of them are a ratio of two things of the same kind.
has one or nottap a chip
Does this have a dimension?
No dimensions. An angle is arc over radius, [L] ÷ [L] = [M⁰L⁰T⁰]. A radian is really a ratio, so it has a unit but no dimension.
No dimensions. μ is frictional force over normal force, a force over a force. The same thing top and bottom, so everything cancels.
No dimensions. The density of a substance over the density of water. Again, a ratio of two things of the same kind.
No dimensions. The ratio of the speed of light in two media, so [LT⁻¹] ÷ [LT⁻¹].
No dimensions. It is only a number. Every constant is [M⁰L⁰T⁰], and that is exactly why a dimensional check cannot see a ½ or a 2π.
No dimensions. Strain is change in length over original length, [L] ÷ [L].
It has one: [ML²T⁻²]. Work is force times displacement, and both of those have dimensions, so the product does too.
It has one: [T⁻¹]. How many times per second, so the power of time is −1. Nothing on top, so M and L are both 0.
It has one: [ML⁻³]. Mass over volume. A mass on top and a length cubed underneath, two different kinds of thing, so nothing cancels.
It has one: [LT⁻²]. This is the trap. People assume g has no dimensions because it is a constant, but g is an acceleration, and acceleration is [LT⁻²]. Being constant and being dimensionless are not the same thing.
All ten. Did the pattern show up? Whenever a quantity is a ratio of two things of the same kind, it has no dimensions.
5. The real job: catching mistakes
The most useful thing dimensions do is check formulas, and the rule is single: both sides of a correct physical equation must have the same dimensions. It follows that you can only add or subtract quantities that share a dimension. You cannot add a time to a velocity, any more than you can add five taka to five kilograms. The rule is called the principle of homogeneity.
There are eight formulas below, some right and some wrong. Tap one and both sides get broken down. Where a formula fails, the term that does not belong turns red.
pass or failtap a formula
The dimension test
The pendulum formula is something you will meet later; it is here only as an example to test. Notice that even without remembering the formula, the dimensions tell you whether the fraction inside has been turned upside down. That is the power of the method: you can catch the mistake without knowing the answer.
check yourselfpick one answer
Test the formula
A student writes s = vt². What does a dimension check give?
Right. The right side is [LT⁻¹] × [T²] = [LT]. The left is [L]. One extra power of time is left over, so the formula is definitely wrong; it should be s = vt.
They do not match. The t² brings [T²], which against the [T⁻¹] in velocity leaves [T¹], and there is no time at all on the left.
[LT⁻¹] is the dimension of v alone. Forget to multiply in the [T²] from t² and this is what you get.
It can be tested, and this is exactly what dimensions are best at. The formula was rejected without a single number being put into it.
6. What dimensions cannot do
Dimensions have three uses: testing whether an equation can be right, converting units between systems, and finding relationships between quantities. But there is a limit, and not knowing it leads to a serious mistake.
s = ut + ½at² and s = ut + at² have exactly the same dimensions, because ½ is [M⁰L⁰T⁰]. Only one of them is correct. So dimensions can reject a formula but they cannot pass one. If the dimensions do not match, the formula is certainly wrong. If they do match, all you know is that one easy way of being wrong has been ruled out.
check yourselfpick one answer
Why it does not always hold
Why is “if the dimensions match, the formula is right” not always true?
Right. s = ut + ½at² and s = ut + at² match perfectly, because ½ is [M⁰L⁰T⁰]. Only one of them is correct.
Dimensions work for every quantity, not just lengths. Besides M, L and T there are I and θ.
That is true, but it is not the reason. Dimensionless quantities are not a problem; the problem is that numerical constants go unseen.
This one is simply false. Changing the unit does not change the dimensions: m, cm and feet are all [L].