2.1 Physical quantities and units
Quantitative versus qualitative
Most observation in physics are quantitative Descriptive observations (or qualitative) are usually imprecise
Figure 1: Qualitative Observations How do you Figure 2: Quantitative Observations. measure artistic What can be measured with the beauty? instruments on an aeroplane?
2.1.1 Physical quantities
A physical quantity is one that can be measured and consists of a magnitude and unit. It is a measurable property whose meaning is precisely defined so that everyone can have the same understanding of the term. The meaning of a physical quantity can be represented by :
Mass A DEFINING EQUATION Density= Volume
A WORD DEFINITION The Density of a substance is the mass per unit volume of the substance
When quoting the measurement of a physical quantity it is essential to state the unit as well as the numerical value
Physical quantities can be classified into two types:
Base / Fundamental quantities are the quantities Base quantity is like the brick – the basic on the basis of which other quantities are building block of a house expressed. Fundamental quantities which cannot be expressed in terms of any other physical quantity. e.g. quantities like length, mass, time, temperature are fundamental quantities
Derived quantities The quantities that are expressed in terms of base quantities are called Derived quantity is like the house that was derived quantities build up from a collection of bricks (basic quantity) 2.1.2 S.I units Unit To measure a quantity, we always compare it with some reference standard. To say that a rope is 10 metres long is to say that it is 10 times as long as an object whose length is defined as 1 metre. Such a standard is called a unit of the quantity. 1
Therefore, unit of a physical quantity is defined as the established standard used for comparison of the given physical quantity. The units in which the fundamental quantities are measured are called fundamental units and the units used to measure derived quantities are called derived units.
Système International Units (SI system units) of fundamental quantities
In earlier days, many systems of units were followed to measure physical quantities. The British system of foot−pound−second or fps system, the Gaussian system of cen metre −gram −second or cgs system, the metre−kilogram−second or the mks system, were the three systems commonly followed. To bring uniformity, the General Conference on Weights and Measures in the year 1960, accepted the SI system of units. Table 1: SI fundamental and supplementary quantities In the SI system of units there are seven fundamental quantities and two supplementary quantities.
It is an intriguing fact that some physical quantities are more fundamental than others and that the most fundamental physical quantities can be defined only in terms of the procedure used to measure them.
Système International Units (SI system units) of derived quantities
Table 2: Derived quantities and their units SI system of units are coherent system of units, in which the units of derived quantities are obtained as multiples or submultiples of certain basic units.
A derived quantity has an equation which links to other quantities.
Units of Time, Length, and Mass: The Second, Meter, and Kilogram
The Second
The SI unit for time, the second (abbreviated s), has a long history. For many years it was defined as 1/86,400 of a mean solar day. More recently, a new standard was
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adopted to gain greater accuracy and to define the second in terms of a non-varying, or constant, physical phenomenon (because the solar day is getting longer due to very gradual slowing of the Earth’s rotation).
Cesium atoms can be made to vibrate in a very steady way, and these vibrations can be readily observed and counted. In 1967 the second was redefined as the time required for 9,192,631,770 of these vibrations.
Figure 3: An atomic clock such as this one uses the
vibrations of cesium atoms to keep time to a precision of better than a microsecond per year
The Meter
The SI unit for length is the meter (abbreviated m); its definition has also changed over time to become more accurate and precise. The meter was first defined in 1791 as 1/10,000,000 of the distance from the equator to the North Pole.
This measurement was improved in 1889 by redefining the meter to be the distance between two engraved lines on a platinum-iridium bar now kept near Paris.
Figure 4: 1791 definition of meter
Figure 5: 1889 definition of meter
By 1960, it had become possible to define the meter even more accurately in terms of the wavelength of light, so it was again redefined as 1,650,763.73 wavelengths of orange light emitted by krypton atoms. In 1983, the meter was given its present definition (partly for greater accuracy) as the distance light travels in a vacuum in 1/ 299,792,458 of a second.
Figure 6: 1983 definition of meter
The Kilogram
The SI unit for mass is the kilogram (abbreviated kg); it is defined to be the mass of a platinum-iridium cylinder kept with the old meter standard at the International Bureau of Weights and Measures near Paris.
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Dimensional analysis
In physics, the word dimension denotes the physical nature of a quantity. The distance between two points, for example, can be measured in centimetres, metres, or kilometres, which are different ways of expressing the dimension of length.
The symbols typically used to specify the Table 3: Dimensions of fundamental quantities dimensions of length, mass, and time are L, M, and T, respectively. In physics, it’s often necessary either to derive a mathematical expression or equation or to check its correctness. A useful procedure for doing this is called dimensional analysis, which makes use of the fact that dimensions can be treated as algebraic quantities. Such quantities can be added Table 4: Dimensional formulae of some derived quantities or subtracted only if they have the same dimensions. It follows that the terms on the opposite sides of an equation must have the same dimensions. If they don’t, the equation is wrong. If they do, the equation is probably correct, except for a possible constant factor.
Example problems on dimensional analysis:
(1) Show that the expression = + , is dimensionally correct, where v and v0 represent velocities, a is acceleration, and t is a time interval.
(2) Determine whether the equation x=vt2 is dimensionally correct? If not, provide a correct expression
Homogeneity of physical equations
A physical equation is homogeneous if quantities on BOTH sides of the equation have the same units.
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Note: Numbers have no units. Some constants have no unit e.g. π. A homogeneous equation may not be physically correct but a physically correct equation is always homogeneous.
Table 5: Prefixes for power of ten Prefixes and their symbols
Magnitudes of physical quantity range from very large to very small. E.g. mass of sun is 1030 kg and mass of electron is 10-31 kg.
Prefix is used to describe these magnitudes.
SI units are part of the metric system. The metric system is convenient for scientific and engineering calculations because the units are categorized by factors of 10.
Convention for labelling tables and graphs v/ms-1 The symbol / unit is indicated at the italics
Then fill in the data with pure numbers
Then plot the graph after labelling x axis and y axis t/s
Estimation
On many occasions, physicists, other scientists, and engineers need to make estimates or “guesstimates” for a particular quantity. What is the distance to a certain destination? What is the approximate density of a given item? About how large a current will there be in a circuit? Many estimates are based on formulae in which the input quantities are known only to a limited accuracy. 5
As we develop problem-solving skills (that can be applied to a variety of fields through a study of physics), we will also develop skills at estimating. These skills are developed through thinking more quantitatively, and by being willing to take risks. As with any endeavour, experience helps, as well as familiarity with units. Estimations allow us to rule out certain scenarios or unrealistic numbers. It also allows us to challenge others and guide us in our approaches to our scientific world. In problem solving or calculations carried out in experiments you should always look at your answer to see if it seems reasonable. When making an estimate, it is only reasonable to give the figure to 1 or at most 2 significant figures since an estimate is not very precise.
Occasionally, students are asked to estimate the area under a graph. The usual method of counting squares within the enclosed area is used.
Example problems on estimations
(1) Can you estimate the height of one of the buildings on your campus, or in your neighbourhood? Estimate the height of a 39-story building.
(2) Using mental math and your understanding of fundamental units, approximate the area of a football ground.
End of section problems
(1)
(2)
(3)
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(4) The speed of sound is measured to be 342 m/s on a certain day. What is this in km/h?
(5) Tectonic plates are large segments of the Earth’s crust that move slowly. Suppose that one such plate has an average speed of 4.0 cm/year. (a) What distance does it move in 1 s at this speed? (b) What is its speed in kilometres per million years?
(6) How many heartbeats are there in a lifetime?
(7) A generation is about one-third of a lifetime. Approximately how many generations have passed since the year 0 AD?
(8) How many times longer than the mean life of an extremely unstable atomic nucleus is the lifetime of a human? (Hint: The lifetime of an unstable atomic nucleus is on the order of 10−22s .)
(9) Calculate the approximate number of atoms in a bacterium. Assume that the average mass of an atom in the bacterium is ten times the mass of a hydrogen atom. (Hint: The mass of a hydrogen atom is on the order of 10−27 kg and the mass of a bacterium is on the order of 10−15 kg. )
(10) Estimate the following:
Bibliography 1. OpenStax College, College Physics. OpenStax College. 21 June 2012.