Units and Measurements

When you learn about units and measurements, you’ll learn the basics of how we can solve our problems in small ways that can have a big impact. Today we’re going to be exploring the metric system and some common measurements you’re probably familiar with.

For example ~ One of the most basic units of measurement in the physical world is the metre. The metre is defined by the distance travelled by light in a vacuum in 1/299,792,458 of a second. In the SI system of measurement, metre is the unit of length.

Units and Measurements:

The first thing you should know about units and measurements is that there are different types, and each has its own uses. Some units are used to express base quantity and some are used to express derived quantities.

Here, we’ll focus on some of the most common units and measurements you probably know.

Ways to write units of physical quantities:

1. Units that are named after scientists should not be written with a capital letter, such as newton, watt, ampere, and metre.
2. The way that you write units should be either in full or in agreed symbols only.
3. Units cannot be written in the plural form. For example- 20 kg but not 20 kgs, 10 w but not 10 wass.
4. Within and at the end of symbols for units, there should be no full stop or punctuation mark. For example – 30 W but not 30 W.

Dimensional analysis

Physical quantities are related to the dimensions of the units of measurement that are used to define them. This helps us perform mathematical calculations that are easier, more precise, and quicker. In other words, it is the study of dimensional formulae. It is the technique used to manipulate dimensional formulae.

Before studying the dimensional formula we should first understand dimensional constants:

The dimensional constants are the physical quantities that have dimensions and have a fixed value.  Examples of the dimensional constant are Planck’s constant (h = 6.63 x 10-34  J s), gravitational constant (G = 6.67 x 10-11 Nm2/kg2), Coloumb force constant (k = 9 x 109 Nm2/C2), etc.

Uses of Dimensional Formula:

1. It can prove to be a helpful tool to check the consistency and coherency of your dimensional equation.
2. The dimensional formula will be used to establish the correlation between the physical quantities of physical phenomena.
3. These formulas can be used to change the units from one system to another.

There are some limitations of dimensional formulas:

1. It is not concerned with the dimensional constant.
2. The formula that contains functions like trigonometric, exponential, logarithmic, and the like cannot be derived.
3. It does not provide any information about whether the quantity is a scalar or vector.

Numerical problems:

1. Q) A physical quantity X is given by the expression X = ab2/c. Calculate the percentage error in quantity X, if the percentage errors of quantities are given as Δa/a = 4 % ,Δb/b = 2% and Δc/c = 3%.

1. A) ΔX/X = Δa/a + 2Δb/b + Δc/c

     ΔX/X = 4/100 + 2×2/100 + 3/100

     ΔX/X = 11/100

         So the percentage error of X is 11%

1. Q) The percentage error in the measurement of the radius and height of the cylinder are represented by x and y. Can you find the percentage error in the measurement of volume? Which measurement needs more attention? 
2. A) Volume is given by: V = πr2h

     Error in volume is given by 

          ΔV/V = 2Δr/r + Δh/h 

    ΔV/V = 2 x/100 + y/100 

     ΔV/V =  (2x + y)/ 100 

Hence, the percentage error is  (2x + y)/ 100.

Conclusion:

In this article, we have learned about the topic units and measurements. We have focused on some of the most common units of measurement and ways to write units of these physical quantities. Dimensional analysis helps us perform mathematical calculations that are easier, more precise, and quicker. It is a method used to manipulate dimensional formulae.