Length, Mass and Time Measurement

In this topic, we will cover the need for measurement and how to measure basic quantities such as length, mass and time and their corresponding units.

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Summary

1. The length of the object varies from 10⁻¹⁵ meters(size of the nucleus) to 10²⁶ meters(size of the universe).
2. The distances ranging from 10⁻⁵ meters to 10² meters can be measured by using direct measurement methods by using a scale.
3. Atomic distances(10⁻¹⁵ meters) and Astronomical distances(10²⁶ meters) are calculated using indirect methods.
4. For measuring long distances following methods can be used
   1. Echo Method
   2. Triangulation Method
   3. Parallax Method
   4. Kepler’s Third Law Method
   5. Copernicus Method
5. The mass of the object varies from 10⁻³⁰ kilograms(mass of an electron) to 10⁵⁵ kilograms(mass of the universe).
6. Mass of the body which is under the effect of gravity is called gravitational mass. It is measured by using physical balance.
7. Mass of the body when it is in a translational motion under the effect of the external force is called as inertial mass. It is measured by using inertial balance.
8. The weight of the body is defined as the gravitational pull of the earth because of its mass. It is measured by using spring balance. Its unit is not kilogram as many think. Its unit is kilogram-force (or newton).
9. Formula: Weight= Mass x gravity
10. The time interval varies from 10⁻²² seconds to 10¹⁸ seconds(life of sun).
11. Any device with uniform speed such as oscillating pendulum can be used to measure time interval.
12. One second is defined as 9 billion oscillations of the cesium atom.
13. Very long time intervals such as the age of rock are measured using radioactive decay method.
14. Small time intervals are measured using an atomic clock.

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We will go through the story of measurement, measurement of length, mass and time. This will help you get a perspective on how things evolved and may aid you in thinking about how to solve problems.

Measurement of Length

Let us start with length now. Before the scales were invented people used to use the width of their hands or length of their arms to measure the length of something right so for example the width of this table can be said to be five hands essentially what we are doing is that we are comparing two lengths isn’t it we are saying how many times is one length in terms of another known length so measurement of any kind is basically a comparison against a known standard and here we are using the width of her hand as a standard but the issue is that everyone’s hand has a different width so as long as we need a crude estimate of length it was okay but when more accurate measurements were required especially in cases like construction of buildings or scientific experiments this kind of crudeness could not be acceptable so how could one use a standard measurement.

Standardisation of Measurement

Let’s see if you can think about how would you go about solving this problem so here is a question for you as most of you have guessed it is the king’s hands or foots length that could be made as standard if you choose yours well many people may not agree right so something similar happened on the 12th or the 13th century when the King of England Henry, once said that in his kingdom the unit of measure of length will be called a yard, which will be the distance from the tip of his nose to the end of his outstretched arm and then, of course, a primary scale was made from this measurement and from this primary scale, secondary scales were manufactured and distributed to everyone for use so now everyone had access to standard way to measure length the King of France – did something similar in France the length of the Royal foot was made a standard for measuring length and this is said to be the original definition of the foot that we used today although it underwent some changes over the years and since modern scientific and mathematical thought mostly evolved in England in France during sixteenth and seventeenth-century.

Conversion of Units

The standards that we use today in the world had their origins in one of these two countries you may also appreciate the fact that with lack of modern communication during those times standards for measurement in countries were usually developed in isolation that’s why we see so many different systems of measurement like the SI system or the MKS system or the British engineering system so with many different types of units doing the round it becomes important that we are able to convert the units from one system of measurement to another for example think about a situation where somebody from England goes to France and meets a local carpenter there because he wants a table made for himself so he tells a carpenter I’d like you to make me a table which is 2.5 yards by one yard so what he means is that the length of the table is 2.5 yards and the width is 1 yard but yard is a unit of length used in England so our French carpenter doesn’t understand what this guy is talking about so he tells him his problem and asks if he could convert yard into foot because that’s what he can measure so the Englishman took a yard and a foot scale and figured how much was a yard – a foot and it turned out to be that there are about approximately 3 feet in a yard so he told the carpenter that he wanted a table which was seven and a half feet by three feet which the carpenter understood and went on to make the table the point is with different types of measurement knowing around in the world it is important to keep in mind how to convert from one type of unit to another.

SI System of Units

The SI system of units is the modern form of the metric system in this system the standard for length or unit measure for length is defined as a meter which is the length of this platinum-iridium rod kept in the International Bureau of weights and measures near Paris as more precision was required the modern definition of metre is slightly changed and is now defined in terms of the wavelength of krypton 86 emissions but we don’t have to get into that the idea of measurement using comparison from some standard should be clear to you now.

Measurement of Mass

The next fundamental physical quantity that we’ll talk about is mass and see how mass is measured now mass is closely related to the amount of matter in a body but by definition it is a measure of inertia of the body, okay and this inertia is slightly different than how Aristotle had defined it earlier so this inertia is that property of matter by whose virtue a body resists any change in its state of motion or rest okay in simple words if a body is at rest we need some effort to put it in motion right and we also need some effort to stop the motion of a body to bring it to rest so more the mass of a body means that if this body is moving, one will need more effort or force to stop it or if this body is at rest when will need more effort to move it but it has been observed that more the amount of matter in a body more is the effort to put it in motion what is more it’s inertia and hence more its mass and less the amount of matter in a body less is the effort to put it in motion that is less its energy and hence less its mass actually mass is directly proportional to the amount of matter so that is the reason why mass is usually used to represent the amount of matter in the body although by definition it is not exactly the same thing. We will use mass a little loosely to represent the amount of matter unless strictly specified otherwise but don’t confuse yourself.

Mass and Density

Mass does not depend on the size of a body a balloon may be much bigger than the small iron balls but the mass of one such iron ball will be more than that of the balloon and that brings us to a new type of interesting physical quantity which is density so density is a measure of how compactly is matter packed in a given volume, for example, the particles of the iron are more densely packed than those on the balloon which are more spread apart so that’s the idea about density. By definition density of a body is its mass per unit volume now what does that mean to understand consider two containers of equal size. Let’s label them as A and B and because their size is equal their volumes must also be equal all right. Let’s fill these containers with sand or gravel as we said that mass is somehow proportional to the amount of matter in the body, the mass of sand in both these containers must be equal because the quantity of sand is equal I mean at least approximately isn’t it. So what about the volume of sand because the sand is contained in these cylinder shaped containers and hence the volume of the sand will also be the same a solid cylinder now using a sand tamper let’s pound the sand in one of the containers all right let’s say container A till we have some more space in this container and then we just top up the empty space in container A with more sand so now which container has more amount of sand of course A right and as we have already seen that more the amount of matter more is its property of inertia and hence more the mass so that means that the mass of sand in container A is more than that in container B what about the volume of sand well that still remains the same right because the volume of the sand is the same as the volume of the containers and because container A has more mass in the same volume the density of sand in container A is more than that in B so density is represented by the Greek letter Rho and is defined as the amount of mass contained in a unit volume. I have seen some people getting confused about you to know why do we say that density is mass per unit volume or mass in a unit volume well actually whether we say that density is mass per unit volume or mass divided by volume this actually means the same thing all right.

Using Force To Measure Mass

Suppose there is a body of mass m and volume V and assuming that this mass M is evenly distributed across the whole volume so if V units of volume contains a mass M then one unit of volume would contain a mass M divided by V right so that is the reason that we can say the density is amount of mass divided by the volume it occupies or it is amount of mass in unit volume now that was a little digression but coming back to mass we see that the mass of sand in container a is more than that in container b. But as we said earlier measurement has some kind of comparison with a known standard so how do we measure mass, we cannot measure the mass directly but there is an indirect way to use the force of gravity we all know that Earth pulls all bodies towards itself with some force due to gravity and of course we will study gravitation in greater detail later but for now it is sufficient to know that the force of gravity exerted by earth on a body is equal to the mass of the body multiplied by acceleration due to gravity G.

Using Gravity To Measure Mass

Now acceleration due to gravity G as you may already know is a constant on the surface of the earth and its values about nine point eight metres per second square so let’s see how we can use the force of gravity to measure the mass of a body someone hundreds or maybe thousands of years back came up with this kind of an old-fashioned weighing scale now if we apply some force on one side of the scale it will tilt on that side right and if we apply an equal amount of force on the other side the scale returns to its original position okay so using the same idea if we put a block of mass m1, where m1 is probably a known mass on one side of the scale, then a force of gravity and when g is applied to that side tilting the scale then if we put a body of mass m2 on the other side a force m2 g will be applied to the other side of this scale if the scale balances out it means that the force of gravity applied to the masses on each side of the scale is equal which means that m1 g is equal to m2 g or m1 is equal to m2 because G is constant so now if mass m1 is a known standard one can measure an equal mass of vegetables by comparison and the unit of Mass in the SI system is called a kilogram and the standard kilogram prototype is kept at the International Bureau of weights and measures near Paris under controlled conditions.

Measurement of Time

The next physical quantity that is interesting in the study of physics is time. There are two kinds of time, the one that you read in a clock for example if somebody asks you what time of the day is it and another kind of time is the duration. For example how long does it take for you to reach school?

In physics we are more interested in duration and not the time as we see in a clock for example how much time does it take for this gentleman to cross the screen so that’s a duration or interval of time between two events the first event is his appearance on the screen and the second event is the event of his disappearance and we may be interested in knowing how much time has passed between these two events so again we need some standard time interval or duration to compare against. In earlier times people used to make use of our glasses to keep track of time duration.

One unit of time would be the duration it took for the sand to completely drain from one side to the other for example the time taken for this person to cross the screen in terms of hourglass time is one, two, three, probably four hourglass times. These glasses are not accurate and today nobody uses them to measure time but you get the idea that we need a standard time duration so that we compare other duration in terms of this standard.

The new standard of time as per SI system is a second a highly accurate atomic clock is kept at National Institute of Standards and Technology near France which acts as a standard for a second and well actually the standard has changed to a newer one which defines the second in terms of the frequencies of radiation of caesium 133 atom but again we don’t have to worry about all those details. So, there is a very accurate clock kept somewhere which tells us how long a time duration is in a second so that other time intervals can be compared and measured in terms of it.

Basic or Fundamental Physical Quantities

For various reasons which are helpful in the study of physics any system of units defines some basic or fundamental physical quantities, for example, the SI system defines these seven quantities as base quantities in another system of units probably these quantities could be different. But in the SI system, they are fixed as seven now. Out of these seven, lengths, mass and time are mostly used in mechanics and we already have some idea about these so we will keep most of our discussion to these three quantities and we will talk about the others as we go along just to avoid overwhelming you with a lot of information so these are the fundamental physical quantities.

Derived Quantities

Consider a rectangle now, we usually call the sides of a rectangle as length and width all right or sometimes it may be called as width and height but which basic physical quantity are we trying to measure here is width and height, some sort of length or you know mass or time well obviously it’s the length right so width and height are just the names given to a side of the square but essentially the type of these physical quantities is the length. After defining the basic physical quantities all the other physical quantities can be derived from them so in any system of units and measurement, all the physical quantities apart from the basic ones are derived physical quantities. For example the volume of say a cuboid which is length into width into height but as we just saw width and height are some kinds of length so volume can be defined as the cube of length. This idea may seem a little new so take some time for it to settle down and you will see the importance of it in the next few sections when we talk about dimensional analysis let us look at another physical quantity density okay which we just saw is defined as mass per unit volume so in terms of the basic quantities density can be written as mass divided by volume which is equal to mass divided by the cube of the length. Length being the base quantity and Mass as we already saw is another base quantity another example of derived quantity is the speed with which is the distance traveled per unit time and distance is a form of length so the speed can be written as length divided by time.

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