Least Count of Vernier Callipers and Screw Gauge Physics Notes

Least count of vernier callipers and screw gauge:
(A) Vernier callipers: It is a precision instrument that can be used to measure internal and external distances accurately.

Construction of vernier callipers:
1. Adjacent figure: A and B represents side jaws. They are used to measure the external diameter or width of an object. Here, A is a fixed Jaw and Bis a movable jaw. C and D represent inside jaws. They are used to measure the internal diameter of an object. Here, C is a fixed jaw and D is a movable jaw.

Vernier Callipers

2. Main scale: The main scale consists of a metalic steel strip graduated with the value of one division on it equal to 1 mm. It is generally 15 cm to 20 cm long. It carries the inner and outer measuring jaws. When the two jaws are in contact, the zero of the main scale and zero of the vernier scale should coincide. If both the zeros do not coinside, there will be a positive or a negative zero error.

3. Vernier scale: Vernier scale is a small movable scale which can slide along the main scale. It has 10 divisions marked on it, such that the total length of 10 divisions is equal to 9 mm. In other words, 10 vernier scale divisions coincide with nine main scale divisions.

Least count: Least count refers to the smallest distance that can be measured using an instrument. It indicates the degree of precision of an instrument.

The least count as the name suggests is the finest measurement you can take with the help of that vernier calliper. The least count (LC) of vernier scale is given as:
Least count = \(\frac{\text { Least count of main scale }}{\text { Number of divisions on vernier scale }}\)

Least count of the main scale: The main scale is callibrated in millimetres. To get the least count of the main scale, count the number of divisions on the main scale in 1 cm of it. Divide 1 cm into that much number of divisions; the value obtained is the least count 0 the mains scale in cm. For example, If there are 10 divisions is 1 cm of the main scale, its least count.
= \(\frac{1}{10}\) = 0.1 cm

Number of divisions on vernier scale: Count the number of divisions on the vernier scale. Use a magnifying glass if necessary. In most vernier caliper, the vernier scale has 10 divisions.
Thus, least count of vernier calliper
= \(\frac{0.1}{10}\) = 0.01 cm
Since the least count of the vernier calliper is 0.01 cm, it can be said that when measuring with the help of this vernier calliper, you can get an accuracy of 0.01 cm. They are extensively used to measure the internal and external diameters of tubes as well as dimensions of many more objects.

(B) The screw gauge: The screw gauge is an instrument used for measuring accurately the diameter of a thin wire or the thickness of a sheet of a metal. It consists of a U-shaped frame fitted with a screwed spindle which is attached to a thimble.

Screw Gauge

Parallel to the axis of the thimble, a scale graduated in mm is engraved, This is called the pitch scale. A sleeve is attached to the life head of the screw.

The head of the screw has a hatchet which avoids undue tightening of the screw. On the thimble, there is a circular scale known as the head scale which is divided into 50 or 100 equal parts.

A stud with a plane end surface calW the anvil is fixed on the ‘17’ frame exactly opposite to “the tip of the screw. When the tip of the screw is in contact with the anvil, usually, the zero of the head scale coincides with the zero of the pitch scale.

Pitch of the screw-gauge: The pitch of the screw-gauge is the distance moved by the spindle per revolution. To find out this, the distance advanced by the head scale over the pitch scale for a definite number of complete rotation of the screw is determined.
The pitch can be represented as:

Pitch of the screw = \(\frac{\text { Distance moved by screw }}{\text { No. of full rotations given }}\)…(1)

Least-count of the screw gauge:
The least count (LC) is the distance moved by the tip of the screw, when the screw is turned through 1 (one) division of the head scale.

The least count can be calculated using the formula:
Least count = \(\frac{\text { Pitch }}{\text { Total no. of divisions on the circular scale }}\) …(2)
Total no. of divisions on the circular scale
If the pitch of the screw is 1 mm and
If circular disc is divided into 100 equal parts,
Then L.C. = \(\frac{1}{100}\) = 0.01 mm = 0.001 cm

Physics Notes

Measurement of Large Distances Physics Notes

Measurement of Large Distances:
Length: It is defined as the difference between two positions taken by two events that occur instantly.
1. Parallax Method: When an object is seen by closing our right and left eye alternatively, there is a shift in the position of the object w.r.t. the background observed. This is known as parallax.

Imagine an object P placed at a distance x from our eyes. Let the line joining the object to left and right eye makes 0 angle w.r.t. each other. 0 is called parallax angle.

The distance LR is called basis.
θ = \(\frac{\text { Length of Arc }}{\text { Radius }}\)

Parallax Method
θ = \(\frac{b}{x}\) or x = \(\frac{b}{\theta}\)

2. Size of Astronomical Object: Diameter of Moon: Let Moon be the astronomical object whose diameter is to be measured. We observed Moon with the help of a telescope. Let it be observed from a place E on Earth and make an angle 0 with the two ends P and Q of moon and the point E, as shown in fig. 1.5. 0 is called the angular diameter of the Moon.

Let d be the distance of the Moon from the Earth.

Then, angular diameter, θ = \(\frac{\overline{P Q}}{d}\)
= \(\frac{\text { Length of the arc }}{\text { radius }}\)
or = \(\frac{D}{d}\)
or D = θd

3. Reflection or Echo Method: A gun is fired towards the hill and the time taken between instant of firing and hearing of echo be t.

In this time interval, sound first travels towards the hill from the place of firing and then back from the hill to the place of firing.
Let v be the speed of sound, x be the distance of hill from the place of firing the gun,
then, 2x = v × t or x = \(\frac{v t}{2}\)

Physics Notes

Uses and Limitations of Dimensional Equations Physics Notes

Uses of Dimensional Equations:
The dimensional equations have the three following uses:

1. For changing the magnitude of a physical quantity from one system of unit to another system of unit.
2. To check the accuracy of physical relation.
3. To establish a relationship i.e., formulae between different physical quantities.

1. Conversion of units of physical quantities from one system to another system: The method of dimensional analysis can be used to obtain the value of the physical quantity in some other system, when its value in one system is given.
As discussed earlier, the measurement of a physical quantity is given by:
Q = nu
If the unit of a physical quantity in a system is Ui, and the numerical value is then:
Q = n₁u₁ …(1)
Similar to the other system if the unit is u₂ and magnitude is n₂ then:
Q = n₂u₂ …(2)
From Eqs. (1) and (2)
n₁u₁ = n₂u₂ …………..(3)
If a,b,c are the dimensions of a physical quantity in mass, length and time, then:
n₁ [M₁^aL₁^bT₁^c] = n₂ [M₂^aL₂^bT₂^c]
Here M₁, L₁, T₁ and M₂, L₂, T₂ are the units of mass, length and time in the two systems, then;
n₁ = n₂\(\left[\frac{\mathrm{M}_{1}}{\mathrm{M}_{2}}\right]^{a}\left[\frac{\mathrm{L}_{1}}{\mathrm{~L}_{2}}\right]^{b}\left[\frac{\mathrm{T}_{1}}{\mathrm{~T}_{2}}\right]^{c}\)
The equation can be used to find out the value of a physical quantity in the second or the new system, when its value in first system is known.

Limitations of Dimensional Equations:
The method of dimensional analysis provides simple and quick solutions to so many physical problems. However, it has a few limitations also:
1. This method does not enable us to determine the value of the constant of proportionality which may be a pure number or a dimensional ratio. The value of the constant has to be determined experimentally or by some other method.

2. This method can not be used to derive the relations, such as S = ut + \(\frac{1}{2}\) at , v² – u² = 2as, etc. by usual method. Such relations are called composite relations. Even while deriving such a relation in parts, it does not tell about the nature of the sign (plus or minus) connecting the various terms in relation.

3. This method can not be used to derive a relation in the cases. Where the trigonometric or exponential functions are involved.
Or
r = \(\left(\frac{M}{2 \pi N \rho}\right)^{1 / 3}\)
The value of r, so calculated is of the order of 10^-10 m.

Physics Notes

Dimensional Formulae and Dimensional Equations Physics Notes

Dimensional Formulae and Dimensional Equations:
The dimensional formula of a physical quantity may be defined as the expression that indicates which of the fundamental units of mass, length, and time enter into the derived unit of that quantity and with what powers.

For example, as deduced above [M⁰L¹T^-1] is the dimensional formula of velocity. It reveals that unit of velocity depends on [L], and [T]. It does not depend upon [M]. Further, unit of velocity varies directly as a unit of length and inversely as a unit of time.

Further if we represent velocity by [v], then [v] = [M⁰ L¹ T^-1]
Is called the dimensional formula of velocity. Thus when a physical quantity is equated to its dimensional formula, what we obtain is the dimensional equation of the physical quantity.

The dimensional formulae of some of the important physical quantities are derived below and are listed in Tables, I, II, III, and IV. The SI units of all these quantities are also given in these tables.

Table-I Dimensional formula of physical quantities from Mechanics:

Table-II Dimensional formulae of physical quantities from Electricity and Magnetism.

Table-III Dimensional formulae and SI units of some special physical quantities

Table-IV Quantities having the same dimensions:

Dimensions	Quantity
[M°L°T-1]	Frequency, angular frequency, angular velocity, velocity gradient, and decay constant.
[M1L2T-2]	Work, internal energy, potential energy, kinetic energy, torque, a moment of force.
[M1L-1T–2]	Pressure, stress, Young’s modulus, energy density.
	Momentum, impulse
[M0L-1T-2]	Acceleration due to gravity, gravitational field intensity
[M1L1T–2]	Thrust, force, weight, energy gradient
[M1T-1T-2]	Angular momentum and Planck’s constant
[M1L0T-2]	Surface tension, energy per unit area
[M0L0T0]	Strain, refractive index, angle, solid angle, relative permittivity, relative permeability.
[M0L2T-2]	Latent heat and gravitational potential
[M0L0T1]	\(\frac{L}{R}\), \(\sqrt{LC}\) , RCwhere L inductance, R = resistance, C = capacitance.

Physics Notes

Dimensions of a Physical Quantity Physics Notes

Dimensions of a Physical Quantity:

The dimensions of a physical quantity is defined as the powers to which the fundamental quantities are raised in order to represent that physical quantity.
The seven fundamental quantities are denoted with square brackets [ ].

For example length is represnted by [L], mass by [M], the dimension of time is described as [T], electric current by [A] and so on.

Consider a physical quantity Q which depends on base quantities like mass, length, time, electric current, the amount of substance and temperature, when they are raised to power a, b, c, d, e and f.Then dimensions of physical quantity Q can be given as:

[Q] = [M^a L^b T^c A^d mol^e k^f]

It is mandatory for us to use [ ] in order to write the dimension of a physical quantity. Look out few examples given below:

1. The volume of a solid is given as the product of length, breadth, and its height. Its dimension is given as:
Volume = Length × Breadth × Height
Volume = [L] × [L] × [L] (as length, breadth 8 heights are length]
Volume = [L]³

As volume does not depend on mass and time, so the power of mass and time will be zero while expressing its dimensions.
The final dimension of volume will be [M]0{L]3|T]0
= [M⁰ L³ T⁰]

2. Velocity = \(\frac{\text { displacement }}{\text { time }}\)
v = \(\frac{\text { [L] }{\text { [T] }}\) = [M⁰ L^-1 T^-1]
Dimension of velocity are 0 in mass, 1 in length and -1 in time i. e., (0, 1, -1).

3 Acceleration = \(\frac{\text { changein velocity }}{\text { timetaken }}\)
= \(\frac{\text { displacement }}{\text { time } \times \text { time }}\)
a = \(\frac{\left[\mathrm{L}^{1}\right]}{\left[\mathrm{T}^{2}\right]}\)
Dimensions of acceleration are (0, 1, – 2).

4. Force:
F = mass × acceleration
= [M¹] × [M⁰ L¹ T^-2]
= [M¹ L¹ T^-2 ]
dimensions of force are (1, 1, – 2).
Note: In dimensional representation, the magnitudes are not considered.

Physics Notes

Rules to write name and symbol for units of S.I. system Physics Notes

Rules to write name and symbol for units of S.I. system:

Following are the most important rules of the SI regarding the writing of quantities, names and symbols of the units; they are valid in general, i. e., also for units that do not belong to the SI.

1. Symbols and prefixes are the same in all languages.

2. Symbols are written in lower case, except when the unit is derived from a proper name.


3. Symbols are never pluralized. e.g. 65 g (not: 65 gs), 15 km (not: 15 kms)

4. An oblique stroke ( / ) is always used with symbols rather than the word, “per”. For example km/h (not: km per hour). However, kilometre per hour (not: kilometre/hour), when writing units in full.

5. The units that are named after scientists, are not written with a capital letter. For example the unit of force named after Issac Newton is written as ‘newton’.

6. The symbols that are named after scientists are written with capital initial letter. For example the unit newton has symbol N.

7. Most single units will be written in one word, e.g., centimetre, millimeter, kilogram, kilowatt (not: centi-metre, kilo-gram etc.).

8. Always place the symbol behind the numeral for example 16.8 km, 650 g, 350 mL.

9. A number and a symbol should never be separated by an adjective.

Some examples of representation of the unit of measurement:

Physics Notes

Supplementary Units Physics Notes

Supplementary Units:

Definitions:
1. Radian: One radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle.

Plane angle dθ = \(\frac{d S}{r}\)
If dS = r, then dQ = 1 rad

2. Steradian: One steradian is the solid angle subtended at the center of a sphere, by that surface which is equal in area to the square of the radius of the sphere.

Solid angle (dΩ) = \(\frac{\text { Normal surfacearea }}{(\text { Radius })^{2}}\)
∴ Solid angle (dΩ) = \(\left(\frac{d A}{r^{2}}\right)\) steradian

If dA = r² then dΩ = 1 steradian
In case of complete surface area, the cubic angle subtained by it
Ω = \(\frac{4 \pi r^{2}}{r^{2}}\) = 4π steradian

Physics Notes

International Definitions of Fundamental Units Physics Notes

International Definitions of Fundamental Units:
1. Metre: 1 meter is defined as the distance travelled by light in vacuum in \(\frac{1}{299792458}\) of a second.

2. Kilogram: General Conference of Weights and Measures (GCWM) defined kilogram as the mass of Platinum Irridium cylinder kept at the Bureau of Weight and Measures at Sevres near Paris, France.

3. Second: One second is equal to the duration of 9,192,631,770 vibrations corresponding the transition between two hyperfine levels of Caesium 133 atom in the ground state.

4. Ampere: 1 ampere is the constant current which when maintained in each of the two straight parallel current-carrying conductors of infinite length of negligible area of cross-section and held one metre apart in vacuum produce a force of 2 × 10⁷ Nm between them.

5. Kelvin: 1 Kelvin is the fraction \(\frac{1}{273.16}\) of the thermodynamical temperature of the triple point of water.

6. Candela: One candela is that luminous intensity in a perpendicular direction of \(\frac{1}{600000}\) m² area of a black body at the freezing point of Platinum under a pressure of 101. 325 N m^-2.

7. Mole: One mole is that amount of substance that contains as many elementary entities as there are atoms in 0.012 kg of pure ₆C¹².

8. The Kelvin unit of thermodynamic temperature is the fraction \(\frac{1}{273.16}\) of the thermodynamic temperature of the triple point of water.

Physics Notes

International system of units (S.I. system of units) Physics Notes

The international system of units (S.I. system of units):
The international system of units (abbreviated as SI, from the French system international (d’ unites) is the modern form of the metric system, and is the most widely used system of measurement.

The S.I is based on the following seven fundamental units and two supplementary units as listed in the table.
Fundamental quantities and their units

Supplementary quantities and their units:

Example of derived quantities:

Abbreviation in power often: In order to write very large and very small quantities compactly, some times we make use of certain prefixes. The following table gives the prefixes, their symbols and their values expressed as power of 10.

Physics Notes

Systems of Units Physics Notes

Systems of Units:
To measure the fundamental physical quantities length, mass and time, we have three systems of units. They are:

1. C.G.S. System
2. M.K.S. System
3. F.P.S. System (British system)

In all these three systems only three physical quantities mass, length and time are considered to be the fundamental quantities.
1. C.G.S. System: This system of units was set up in France and it is based on centimeter,, gram and second as the three basic units for length, mass and time respectively.

2. M.K.S. System: This system is based on metre, kilogram and second as the fundamental units of length, mass and time respectively. It is also a metric system of units and closely related to the C.G.S. system of units.

3. F.P.S. System: This system of units, also known as the British system of units, is based on foot, pound and second as the fundamental units of length, mass and time. Its use in scientific work is declining more and more.

Merits of S.I. system:
The merits of the international system of units are:

1. This system makes use of only one unit for one physical quantity. Therefore, it is a rational system.
2. S.I is a coherent system of units i.e., a system based on a certain set of fundamental units, from which all derived units are obtained by multiplication or division without introducing the numerical factors.
3. It is a metric system i.e., the multiple and submultiple of units are expressed as a power of 10.
4. It is an internationally accepted system of units.
5. S.I. has a broader base. It has seven base units and two supplementary units.

Physics Notes

Units for measurements Physics Notes

Units for measurements
Measuring any physical quantity means comparing it with a standard to determine its relationship with the standard of the same kind.

The chosen standard of the same kind taken as the reference in order to measure a physical quantity is called the unit of that quantity.

The process of measurement of a physical quantity involves:

1. The selection of unit and
2. To find out the number of times that unit is contained in the physical quantity.

For example, if we are asked to measure the length of a rope and select meter as the unit of measurement. We place the metering rod successively along PQ and find out that it is contained four times in PQ (figure). Thus, 4 is the numerical value of the length PQ.

So, we write PQ = 4 meters.

In general,
The magnitude of a physical quantity Q = numerical value × size of its unit
If unit V of the quantity is contained n times in the quantity, we write
Q = nu

We know that if the size of the chosen unit is small, then the numerical value of the quantity will be large and vice-versa. It is obvious that the measure of the physical quantity is always the same
nu = constant

If n₁ is the numerical value of the physical quantity for unit u₁ and n₂ for unit u₂, then
n₁u₁ = n₂u₂

Characteristics of a standard unit:
The unit chosen for measuring any physical quantity should fulfill the following requirements:

1. It should be well defined.
2. It should be of suitable size i.e., neither too large nor too small in comparison to the physical quantity to be measured.
3. It should be easily accessible.
4. It should be reproducible.
5. It should not change with time and
6. It should not change with the changing physical conditions like temperature, pressure, etc.

Physics Notes