Are you a struggling physics high school student who is having a hard time understanding the concepts of derived quantities and base SI units? This guide provides you with a vocabulary list and definitions of terms, an example of how to find a derived quantity, and some exercises for you to practice.
Vocabulary of Terms
A physical quantity is expressed in units. Units are an extremely important part of a physical quantity. Any measurement without a unit is like a letter without an address. If you master the concepts of units, you will prevent from making mistakes later in your study.
A unit is a standard against which physical quantities can be measured.
A system of units like the SI system defines the units of certain quantities. These quantities are called the base quantities. Here is a list of the base SI units. All other physical quantities can be derived from these base SI units.
A Derived physical quantity is one which can be expressed in terms of the base quantities.
Speed of a body is given by the distance travelled by it per unit time.
Acceleration is the change in the speed of a body per unit time.
An Example of a Derived Quantity
The S.I. system says time and length are base quantities. Hence, “speed = length/time” is a derived physical quantity since it can be expressed in terms of the two base quantities - time and length.
Now, if I define a hypothetical system in which, instead of time and length, the base quantities are speed and acceleration, then since
acceleration = speed/ time, or, rearranging,
time = speed/acceleration
Hence, time becomes a derived quantity for my hypothetical system of units.
Finding the Units of a Derived Quantity
After having identified the derived physical quantity, the next step is to use the expression of this derived quantity in terms of base quantities to derive its units.
Let’s study this by finding the units of acceleration.
Step 1: Identify the derived quantity and express it in terms of the base quantities.
Our derived physical quantity in the example above is Acceleration.
Acceleration = speed/time
= (Distance/time)/time , since speed = distance/time
Rearranging and using length in place of distance,
Acceleration = length/time2
Note that in this type of representation, the magnitudes are not considered. It is the quality of the type of the physical quantity that matters. Thus, initial velocity, a change in velocity, average velocity, final velocity, and speed are all equivalent in this context and will be expressed as length/time.
Step 2: Write down the SI base units of the base quantities used in the equation above.
Using the SI system,
Mass - Kg (Kilogram)
Length - m (metre)
Time - s (second)
Step 3: To find the units of the derived quantity, replace the base quantities by their units.
Unit of acceleration = m/s2
We can express a physical quantity in terms of a product of the base quantities, this is called the dimensional formula.
In the above example,the dimensional formula of acceleration can be written as :
[acceleration] = [LT -2]
T(time) raised to power -2 here means that acceleration depends on the square of time in the denominator.
Here M = Mass, L = Length and T = Time
Whenever you know the dimensional formula of a derived quantity in terms of the base quantities, you can easily find the units as outlined above.
Attempt these questions to test your understanding of the above concepts.
1. Work out the dimensional formulae for momentum, density & displacement.
Given : Momentum = Mass X Velocity; Density is defined as the mass per unit volume; mDisplacement is defined as the shortest distance between two points.
The next one is more of an advanced problem. Good luck!
2. If a hypothetical system of units has work(W), force(F) and acceleration(a) as the base units,
then, it can be seen that [Length] = [W/F] & [Mass] = [F/a]
Is time a derived quantity? How can it be expressed in terms of these hypothetical base units?
Hint : [Work] = [ML2T -2]; [Force] = [MLT -2]; [acceleration] = [LT -2]
Don’t worry about these derivations, just substitute these formulae in the equations given above. ]
Hopefully, these problems will help you get the physics help you need to succeed in your class.
This post is part of the series: Units & Dimensions Study Guide for High School Physics
A resourceful set of articles on units that will come in handy in physics class, as well as using units and dimensions to verify or derive formulae!