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Please Explain Measurement of Area, Volume and Density (Class 11 – Physics).
Please Explain Length Measurement (Class 11 – Physics).
Previously, length was measured using units such as the length of a foot, the breadth of a palm, and so on. The ‘Cubit’ was one of the first means of measuring length. It is the length of the arm from the elbow to the tip of the fingers. These units vary from person to person, resulting in non-unifoRead more
Previously, length was measured using units such as the length of a foot, the breadth of a palm, and so on. The ‘Cubit’ was one of the first means of measuring length. It is the length of the arm from the elbow to the tip of the fingers. These units vary from person to person, resulting in non-uniform measures.
How can we know how far the moon is from the earth or how far the moon is from the sun? How did we determine the earth’s diameter? Measuring length isn’t always simple or easy. We’ll try to respond to these queries in the sections below. In addition, we will learn about the many methods for measuring length.
The measurement or amount of anything from one end to other is referred to as length.
In other terms, it is the largest of the two or the highest of three geometrical form or item dimensions. The width and length of a rectangle, for instance, are its dimensions. Furthermore, under the International System of Quantities, length is a quantity with the dimension distance.
The meter, abbreviated as m, is the basic unit for length in the International System of Units (SI). The length or distance is measured in kilometers (km), meters (m), decimeters (dm), centimeters (cm), and millimeters (mm) in the metric system (mm). It is possible to convert quantities from meters to centimeters, kilometers to meters, centimeters to millimeters, and so on.
There were no modes of transportation available in ancient times. People used to travel on foot or by using animals to transport goods. Over time, the term “wheel” was coined. This signified a significant shift in human forms of transportation. Since then, new forms of transportation have been invented and improved regularly. The steam engine was created, and it had a great influence and was instrumental in shaping the world as we know it today.
As a result, transportation has a lengthy history. Did the folks have any idea how far they had to go? To go to any location, one must first determine how far away it is. This aids in deciding whether to walk, take the train, bus, or fly to that location. To determine how far apart two locations are, we must first determine the distance between them. But what exactly does measurement imply? What is the best way to measure a physical quantity? The comparing of an unknown amount to a known amount is known as measurement. A numerical number known as “magnitude” and a “unit” are used to indicate the outcome of the measurement. A ‘unit’ is a pre-determined unit of comparison against which other physical quantities are measured.
The length of the foot, the breadth of the palm, and other such quantities were used to measure the length in the past. The ‘Cubit’ was one of the first means of measuring length. It is the length of an arm measured from elbow to tips of fingers. These units varied from person to person, resulting in non-uniform measures. A set of standard units of measurement has been recognised all around the world to preserve uniformity in measurements.
The International System of Units (SI) is one of the most widely used measurement systems in the world. The basic unit of length in the SI system is the meter. The C.G.S. system is another system of units in which the centimeter is the basic unit of length.
Historically, the human body served as the foundation for length units.
A foot is 12 inches long, and a yard is three feet long. It was simple to describe how distant the next village was and to determine if an object would fit through a doorway using these dimensions. These dimensions also made it easier for individuals to swap garments and wood.
Scale
Let’s take a closer look at what the triangulation approach entails. How might triangulation assist us in determining the distances between distant stars? The parallax approach makes use of the fact that a triangle may be entirely defined using only three parts. Triangulation is the process of determining the values of a triangle to determine an item’s position. Surveyors and architects frequently employ such techniques.
Triangulation is the method of identifying the location of a point by calculating the angles to it from two known sites rather than calculating distances directly.
Triangulation Example
Let’s put this into practice with a real-life scenario. How can we estimate a large object distance from any distance without actually measuring it? It may be measured using the triangulation method.
Triangulation Method of measurement
The displacement or shift in the apparent location of an item when observed from two distinct points of view is known as parallax.
The two places of vision each have their own line of sight, and parallax is defined as half the angle between them. When you’re driving in a car and glance about, you’ll notice that items far away appear to move more slowly than items closer to you. This is the parallax effect. Because the parallax of nearby objects is greater than that of distant ones, the parallax may be utilised to measure distances.
When the phenomena of parallax is coupled with triangulation, the position of the item may be determined with great precision. The parallax method is commonly used by astronomers to determine the distances between stars.
Distance Measurement by Parallax Method
The principle of triangulation is used to the measuring of distance through parallax. We learned from triangulation that a triangle may be completely specified if two angles and sides are known.
The distance of a faraway star is being computed in the image below. The star closer to Earth than the farthest one gives the limited parallax value. By observing the star from two known places on Earth that form the triangle’s baseline, we may determine the value of the parallax angle.
Parallax Method
Let’s denote the parallax half-angle between two places on Earth ‘p.’ The radius of the Earth is the greatest value of ‘d,’ and the distance of the star may be considered to be just slightly more than that of the sun. Because the distance from the sun is several orders of magnitude lower than the radius of the Earth, the parallax angle we obtain is exceedingly modest.
Application
The distance to an object measured in parsecs (in terms of light speed) is equal to the reciprocal of parallax angle measured in arcseconds.
Relation between the distance of a star, and its parallax is given as:
D = 1 ⁄ p
where D is the distance of star and p is the parallax angle.
To solve the difficulty of tiny ratios, the parallax of a star is most commonly estimated using yearly parallax, which is defined as the difference in a star’s location as seen from the Earth and the Sun. Instead of using the Earth’s radius as a fixed baseline, the radius of the Earth’s revolution around the Sun is used, which increases the size of the baseline and hence the top angle, making it simpler to measure.
However, for any celestial objects near to the Earth, we can consider the diameter of the Earth as a baseline, and the distance of any celestial objects is given as:
x = b ⁄ θ
where x is the distance of the object from the Earth, b is the baseline or diameter of the Earth and θ is the angle subtended by the object.
Problem 1: If a person covers 1.5 yards in one step, how much distance will he cover in 30 steps?
Solution:
Given:
Total number of steps, n = 30
Value of 1 step, d = 1.5 yards
Total distance covered by the person, D = n d
= 30 × 1.5 yard
= 45 yards
Hence, the distance covered by the person is 45 yards.
Problem 2: Astronomers apply which method to determine how far away a star is?
Answer:
Astronomers use parallax to calculate the distance between stars. Trigonometric parallax is another name for parallax.
Problem 3: What is parallax?
Answer:
The two items appear to be coincident when seen in a straight line. There is a relative displacement between the things if they are at separate locations and the eye is shifted sideways. The closer item travels in the opposite direction from the eye, whereas the further object travels in the same direction.
When two things are perceived in a straight line and the eye is shifted to the side, this is referred to as parallax.
Problem 4: What was the conventional method of measuring the length?
Answer:
The length was measured in history with the help of human body. These were based on the several methods like distance from tip of middle finger to bottom of elbow, or human paces or human heights, etc. However, it was discarded later because these methods were different for different countries and were limited to measure the long distances.
Problem 5: The Moon subtends an angle of 1° 55’ at the baseline equal to the diameter of the Earth. What is the distance of the Moon from the Earth? (Radius of the Earth is 6.4 × 106 m)
Solution:
See lessGiven:
The angle subtended by moon, θ = 1° 55’ = 115’
We know, 1’ = 60’’ and 1’’ = 4.85 × 10-6 rad
Therefore, 115’ = (115 × 60)’’ × 4.85 × 10-6 rad = 3.34 × 10-2 rad
The baseline for the Moon is the diameter of the Earth, b = 2 × 6.4 × 106 m = 1.28 × 107 m
Distance of the Moon from the Earth, x = b ⁄ θ
= 1.28 × 107 m ⁄ 3.34 × 10-2 rad
= 3.83 × 108 m
Hence, the distance of the Moon from the Earth is 3.83 × 108 m.
Please Explain System of Units (Class 11 – Physics).
Measurement forms the fundamental principle to various other branches of science, that is, construction and engineering services. Measurement is defined as the action of associating numerical with their possible physical quantities and phenomena. Measurements find a role in everyday activities to aRead more
Measurement forms the fundamental principle to various other branches of science, that is, construction and engineering services. Measurement is defined as the action of associating numerical with their possible physical quantities and phenomena. Measurements find a role in everyday activities to a large extent. Therefore, it is necessary to study and explore the associated elements along with their theoretical foundations, conditions as well as limitations. It defines the units to be chosen for the measurement of various commodities. It also caters to the comparison of plausible units with the ones already existing of a similar kind.
Measurement defined the new standards as well as form transductions for the quantities which do not have any possible access for direct comparison. These physical quantities can be converted into analogous measurement signals.
Measurements may be made by unaided human senses, generally termed as estimates. It can also be estimated by the use of instruments, which may range in complexity from simple rules for measuring lengths to highly complex analogous systems to handle and design the commodities beyond the capabilities of the senses. Thus, the measurements may range from buying some quantity of milk (in L) or to the highly complex mechanisms, such as radio waves from a distant star or the nuclear bomb radiations. Therefore, we can consider that a measurement, always involves a transfer of energy or interaction between the object and the observer or observing instrument.
Measurement of Height of a person
Unit
The unit of a specified physical quantity can be considered as an arbitrarily chosen standard that can be used to estimate the quantities belonging to similar measurements. The units are well accepted and recognized by the people and well within all guidelines.
A physical quantity is measured in terms of the chosen standards of measurement.
The chosen standard is recognized as the unit of that corresponding physical quantity. A standard unit, in short, is a definite amount of a physical quantity. These standard units can be quickly reproduced to create a wide variety of units and are internationally accepted and accessible.
The measurement of any physical quantity is based on a formula, nu,
where, n = numerical value of the measure of the quantity,
u = unit of the quantity.
Standard
The actual physical embodiment of the unit of a physical quantity is termed as a standard of that physical quantity. The standard is expressed in terms of the numerical value (n) and the unit (μ).
Measurement of physical quantity = Numerical value × Unit
For example: Length of a rod = 12 m. Here 12 is its numerical segment and m (meter) is the unit.
Fundamental Units
Fundamental units are elementary in nature, that is, they can be expressed independently without any dependence on any other physical quantity. This implies that it is not possible to resolve it further in terms of any other physical quantity. It is also termed as a basic physical quantity. Fundamental quantities have their own values and units.
| Fundamental Quantities | Fundamental Units | Symbol |
| Length | meter | m |
| Mass | kilogram | kg |
| Time | second | s |
| Temperature | kelvin | k |
| Electric current | ampere | A |
| Luminous intensity | candela | cd |
| Amount of substance | mole | mol |
Supplementary Fundamental Units
There are two other supplementary fundamental units, namely Radian and steradian are two supplementary which measures plane angle and solid angle respectively.
| Supplementary Fundamental Quantities | Supplementary Unit |
| Plane angle | radian |
| Solid angle | steradian |
θ = 1 radian
Ω = 1 steradian
Properties of Fundamental Units
Any standard unit should have the following two properties:
The seven fundamental units of S.I. have been defined as under.
square meter of a full radiator at the temperature of freezing platinum under a pressure of 101325 newtons per square meter.
C.Derived units
The derived units are in usage for the commodities where the units are obtained from a combination of fundamental units. Derived units are sometimes assigned names. For instance, the S.I unit of force is kg ms-2 , termed as Newton (N). The unit of power is kg m2 s-3 , termed as watt (W).
Steps to find Derived Units
Example: Compute the unit of velocity.
Since, we know velocity is a derived quantity, obtained from distance and time(fundamental quantities).
Mathematically ,
velocity = displacement/time
S.I. unit of velocity = 
= m/s
Thus S.I. unit of velocity is m/s.
Some Important derived units
Some of the derived units have been given specific names, depending on the increase in their usage , though they are not recognized in S.I units.
Systems of Units
Any system of units contains the entire set of both fundamental as well as derived units, for all kinds of physical quantities. The preferred system of units are the following :
Note:
While computation of values for any physical quantity, the units for the involved derived quantities are treated as algebraic quantities till the desired units are obtained.
Advantages of S.I Unit System
The S.I unit of measurement is preferred over other units of measurement, because,
Other Important Units of Length
The distances can be infinitely larger in magnitude, which cannot be depicted in terms of meters or kilometers. For instance, the distances of planets and stars etc. Therefore, it is necessary to use some larger units of length such as ‘astronomical unit’, ‘light year’, parsec’ etc. while making such calculations, some of which are :
Problem 1. Convert the unit of G, which is gravitational constant, G = 6.67 x 10-11Nm2/kg2 in CGS system.
Solution:
Since, we have
G = 6.67 x 10-11 Nm2/kg2
Converting kg into grams, 1 kg = 1000 gms
= 6.67 x 10-11 x 108 x 103 cm3/g1 s2
= 6.67 x 108 cm3/g1 s2
Problem 2. Name the S.I units of the following commodities :
a. Pressure
b. Solid angle
c. Luminous intensity.
Solution:
a. Pascal
b. Steradian
c. Candela
Problem 3. Derive the S.I unit of latent heat.
Solution:
Latent heat =
Problem 4: How are A0 and A.U related?
Solution:
Describing both quantities in terms of meters,
Ao = 10-10m
and 1 A.U. = 1.4961011m.
Therefore,
1 A.U. = 1.496 x 1011 x 1010 A0
1 A.U = 1.496 x 1021 A0
Problem 5: Describe 1 light-year in meters.
Solution:
See lessA light-year is a distance travelled by light in 1 year with the speed of light :
= 9.46 x 1011 m
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When Humans look at their childhood pictures, the first thing they realize is how tall or heavy they have become as compared to the early stages of their lives. Noticing the increment in the weight or becoming taller is done through measurement. Measurement is required everywhere in order to specifiRead more
When Humans look at their childhood pictures, the first thing they realize is how tall or heavy they have become as compared to the early stages of their lives. Noticing the increment in the weight or becoming taller is done through measurement. Measurement is required everywhere in order to specifically compare a quantity’s value from its original value. There can be many examples where measurement is required, For example, a thermometer is used to measure the temperature in Celsius of the body, the clocks on the wall are used to measure time in hours, and so on.
Measurement
Calculating the value of one quantity by comparing it with the standard value of the same physical quantity is called Measurement. It can be said that measurement associates the physical quantity with its numeric value.
Normally, objects are measured by placing them next to each other, and it can be explained which one is heavier or taller, etc. Measuring an object gives two results- value and unit of the quantity. For example, The length of the scale is 15 cm long where 15 is the value and the centimeter is the unit used for length.
Metric System
Metric system is the system that came from the decimal system and this system is used to measure basic quantities, metric system paves the way for the conversion of units, that is, the conversion of bigger units into smaller and vice-versa is possible due to the system. The basic quantities that are present are meter, gram, and liters, and they are used to measuring quantities of length, volume/mass, and capacity.
Measurement of Length
According to the metric system, length is calculated in meters. However, it can easily be converted into other forms depending upon the requirement, for instance, if the requirement is to measure the bigger quantity, it should be measured in kilometers, if the requirement is to measure something smaller, it should be in centimeters. Let’s say, the requirement is to measure a certain distance, it should either be in meters or kilometers.
Below is the easier way to explain the conversion,
Measurement of Area
In the standard way, the area of any quantity is measured in meter2, since the area is a two-dimensional quantity that is scalar in nature. It involves lengths going in two different directions known as length and breadth. The area of bigger and smaller quantities can be converted easily using different units, for example, if the area of a small table is to be measured, it is measured in cm2 and if the area of the plot is to be measured, the unit used is meter2.
Below is the easier way to explain the conversion,
Measurement of Volume
The volume of any quantity is three-dimensional in nature, that is, the length if going in three directions, unlike length or area, the volume contains capacity. The standard unit used to measure volume is meter3. The unit is converted into bigger and smaller units like decimeter3 or kilometer3 based on how big or small the quantity is, it is done by simply dividing/ multiplying by tens, hundreds, thousands, and so on.
Below is the easier way to explain the conversion,
Measurement of Density
The density of any object is defined as the mass of that object per unit volume. Density tells how close or far away molecules are packed in a certain volume. The very famous scientist known as Archimedes discovered the concept of the Density of an object. In the metric system, Density is measured in kg/m3 and is represented as D or ρ. Therefore, it can be denoted as,
Sample Problems
Question 1: Convert the following: m3 to mm3, liter to meter3, mile3 to km3.
Solution:
Question 2: What is the difference between the metric system and the imperial system of measurement?
Answer:
Question 3: Calculate the density of an object having a mass of 1200kg and its volume is 10m3.
Solution:
Question 4: There are two large boxes filled with biscuits. The first has 10 biscuits and the second has 20 biscuit packets present in it. The box have the same volume. Explain which box will weigh more?
Answer:
Question 5: A cube is given which has a volume of 1000m3. Calculate the surface area of the cube in cm3.
Solution:
Question 6: The length and the breadth of a cuboid are same, but the height is twice in value. If the volume of the cuboid is 54000m3, find the length, breadth and height of the cuboid in centimeters.
Solution:
Question 7: A cm scale has a limit of 15 points, how long is the scale in meters?
Solution:
See less