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Units and Vectors

Unit Mistake pays Huge Dividends. The importance of Units In December 1998, NASA launched the 125–million dollar first weather satellite, Mars Climate Orbiter. While flight computers on the ground did calculations based on pounds of thrust per second (an English unit), the spacecraft's computer used Newtons (a Metric unit). A check to make sure the values were compatible was never done. One Pound of force is equal to 4.45 N, but the spacecraft computer at NASA treated it to be 1 N. After a 286–day journey from Earth the Orbiter crashed due to smaller engine thrust, resulting its placement in a lower orbit. The engines fired, but the spacecraft fell too far into the planet’s atmosphere causing it to crash on Mars. Thus the units for any measurement are most important. Let's learn more about the units and measurements of physical quantities.

Learning Objectives

After completing the topic, the student will be able to:

  • Understand the importance of fundamental units of physical quantities and distinguish between the two systems of units.
  • Explore the rules for writing the units for any physical quantity.
  • Estimate and measure very small and very large distances.
  • Determine the dimensions of any physical quantity, using dimensional analysis.
  • Understand the nature of errors and methods of analysis of the errors in measurement.
  • Observe and understand the difference between accuracy and precision.
  • Determine the number of significant figures in a numerical value.
  • Analyse the least count of some measuring devices
  • Differentiate between vectors and scalars and investigate the importance of vectors with relevance to daily life applications.
  • Express vectors in different types of co-ordinate system.
  • Identify the types of vectors and resolve them in to their corresponding components, based on the angle subtended.
  • Employ methods for addition and multiplication of vectors, with reference to the properties of dot and cross products.
length mass and time Length, Mass and Time In general, the three observable Physical quantities Length Mass and Time that can be measured directly are named as the fundamental physical quantities through which others physical quantities can be derived. All measurements are necessarily approximations, a great deal of effort must be taken to make measurements as accurate as possible.
Units for Measurement

Einstein (1933) has said, "Pure logical thinking cannot yield us any knowledge of the empirical world; all knowledge starts from experience and ends in it. Propositions arrived at by purely logical means are completely empty as regards reality". Standard measurements as part of experimentation is essential part of research as well as application of knowledge.

When you go to buy sugar, you do not say that "I want to buy a sugar", you will say "I want to buy 1kg of sugar". The shopkeeper then weighs the sugar, against STANDARD weights and gives you 1 kg of sugar. Similarly, when you measure with a ruler, say length of a pencil, the markings on the ruler are STANDARD lengths. Take another physical quantity, say time. A second, a minute and an hour on your watch has been set to a STANDARD. Now you may ask, why this is so? The answer to this question is that anywhere on the earth, your measurements should match anybody else's measurements.

When you say that you have bought 10 kg of potatoes, whether you buy them in Pune, Paris or Portsmouth, 10 kg of potatoes have to be 10 kg of potatoes and nothing else. Therefore, whenever you measure any physical quantity, they are relative to or compared against some STANDARD measurements. The international community from time to time announces new, revised standards and all nations have to comply with these standards.

The basic quantity of the standard is known as a unit. The amount of a physical quantity that is used as a reference for the measurement of that quantity is called the unit of that quantity. The unit is invariable, easily reproducible and is internationally accepted while an actual measurement would be a fixed multiple of the unit. The multiple is called the magnitude of the unit. For example, if we decide unit length to be 1 meter, and if a length of the side of the table is 2.0 meters then magnitude of the measurement is 2.0.

Meter Historical Standard Platinum-Iridium meter bar The definition of the meter (m), which is the international unit of length, was once defined by a physical artifact – two marks inscribed on a platinum-iridium bar.
Fundamental and Supplementary units
System of Units

Since the development of measuring processes, a number of systems of unit have been put forward. Here are a few:

  • F P S System (Foot–Pound–Second)
    In this system of unit, length is measured in foot; mass in pound; and time in second. This is also known as British system of measurement.
  • C G S system (Centimeter–Gram–Second)
    In this system of unit, length is measured in centimeter; mass in gram; and time in second.
  • M K S system (Meter–Kilogram–Second) or Metric system
    Here, length is measured in meter; mass in kilogram; and time in second.
  • SI Units
    The representations or prototypes of units are maintained in many international centres worldwide, like the International Bureau of Weights and Measures at Sevres near Paris, National Institute of Standard Technology or National Physical Laboratory in the USA or Bureau of Indian Standards in New Delhi, India. Internationally everyone strictly follows a standard system of units. This system is called the S.I. units (Standard International). Calibrations used for measurement have to be standardized so that they remain constant, under normal everyday life. If this was not done, then there would be chaos in the world as everyone's measurement would be different!! In SI there are seven fundamental units and two supplementary units.

MKS units and the SI system of units are nearly similar and now-a-days used in most measurements.

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