Units and Measurements Class 11 Notes: CBSE Physics Chapter 1

Units:

A unit can be defined as an internationally accepted standard for measuring quantities.

• Measurement has been included in a numeric quantity along with a specific unit.

• The units in the case of base quantities (such as length, mass etc.) are defined as Fundamental units.

• Derived units are the units that are the combination of fundamental units. 

• Fundamental and Derived units constitute together as a System of Units.

• An internationally accepted system of units can be defined as Système Internationale d’ Unites (This is how the International System of Units is represented in French) or SI. In 1971, it was produced and recommended by the General Conference on Weights and Measures.

• The table shown below is the list of 7 base units mentioned by SI.

There are two units along with it. They are, radian or rad (unit for plane angle) and steradian or sr (unit for solid angle). Both of these are dimensionless.

Significant Figures

Every measurement gives us an output in a number that includes of reliable digits and uncertain digits.

Reliable digits added with the first uncertain digit can be defined as significant digits or significant figures. This represents the precision of measurement which is dependent on the least count of instrument used for measurement.

The period of oscillation of a pendulum is 1.62 s can be taken as an example. Here 1 and 6 will be reliable and 2 is uncertain. Hence, the measured value will have three significant figures.

Rules for the determination of the number of significant figures

• All non-zero digits will be significant.

• Irrespective of the decimal place, all zeros between two non-zero digits will be significant irrespective of the decimal place.

• Zeroes before non-zero digits and after decimal are not considered as significant, for a value less than 1. Zero presents before the decimal place in case these numbers will be insignificant always.

• Trailing zeroes in case of a number without any decimal place will be insignificant.

• Trailing zeroes in case of a number with a decimal place will be significant.

Cautions for removing ambiguities in calculating the number of significant figures

• Variation of units will not change number of significant digits. As an example, 

$\text{4}\text{.700 m=470}\text{.0 cm}$ 

$\text{              =4700 mm}$ 

Here first two quantities have 4 but the third quantity is having 2 significant figures.

• Make use of scientific notation for reporting measurements. Numbers must be shown in powers of 10 such as \[\text{a }\!\!\times\!\!\text{ 10b}\]where b is defined as the order of magnitude. Example,

$\text{4}\text{.700 m = 4}\text{.700  }\!\!\times\!\!\text{  1}{{\text{0}}^{\text{2}}}\text{ cm }$

$\text{               = 4}\text{.700  }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{3}}}\text{ mm }$

$\text{               = 4}\text{.700  }\!\!\times\!\!\text{  1}{{\text{0}}^{\text{-3}}}\text{ km}$

Here, as the power of 10 is being irrelevant, the number of significant figures will be 4.

• Multiplying or dividing exact numbers will give an infinite number of significant digits. Example,\[\text{radius=}\frac{\text{diameter}}{\text{2}}\]. In this case, 2 can be represented as 2, 2.0, 2.00, 2.000 and so on.

Rules for Arithmetic operation with Significant Figures

Rules for Rounding off the uncertain digits

Rounding off will be essential for reducing the number of insignificant figures to hold to the rules of arithmetic operation with significant figures.

Rules for the determination of uncertainty in the results of arithmetic calculations

For calculating the uncertainty, the below process must be used.

• Do a summation of the lowest amount of uncertainty in the original numbers. Example uncertainty for 3.2 will be \[\pm 0.1\] and for 3.22 will be \[\pm 0.01\].

• Find out these in percentage also.

• The uncertainties get multiplied/divided/added/subtracted after the calculations.

• In the uncertainty, round off the decimal place to obtain the end uncertainty result.

For example, for a rectangle, 

Suppose length,  \[\text{l=16}\text{.2 cm}\] and breadth, \[\text{b=10}\text{.1 cm}\]

After that, take \[\text{l=16}\text{.2}\pm \text{0}\text{.1 cm}\]or \[\text{l=16}\text{.2 cm}\pm \text{0}\text{.6  }\!\!%\!\!\text{ }\] and

breadth \[\text{=10}\text{.1 }\!\!\pm\!\!\text{ 0}\text{.1 cm}\]or \[\text{10}\text{.1 cm }\!\!\pm\!\!\text{ 1  }\!\!%\!\!\text{ }\]

When we multiply,

\[\text{area=length }\!\!\times\!\!\text{ breadth=163}\text{.62 c}{{\text{m}}^{\text{2}}}\text{ }\!\!\pm\!\!\text{ 1}\text{.6  }\!\!%\!\!\text{ }\]

Or \[\text{163}\text{.62}\pm \text{2}\text{.6 c}{{\text{m}}^{\text{2}}}\]

Hence after rounding off, area\[\text{=164}\pm \text{3 c}{{\text{m}}^{\text{2}}}\].

Therefore \[\text{3 c}{{\text{m}}^{\text{2}}}\]will be the uncertainty or the error in estimation.

Rules

1. In the case of a set of experimental data of ‘n’ significant figures, the result must be accurate to ‘n’ significant figures or less (only in the case of subtraction).

For example \[\text{12}\text{.9-7}\text{.06=5}\text{.84 or 5}\text{.8}\](when we round off to least number of decimal places of original number).

2. The relative error of a value of number mentioned to significant figures will be dependent on n and on the number itself.

As an example, say the accuracy for two numbers 1.02 and 9.89 be \[\pm \text{0}\text{.01}\]. But relative errors are:

For \[\text{1}\text{.02,}\left( \frac{\text{ }\!\!\pm\!\!\text{ 0}\text{.01}}{\text{1}\text{.02}} \right)\text{ }\!\!\times\!\!\text{ 100  }\!\!%\!\!\text{ =}\pm \text{1  }\!\!%\!\!\text{ }\]

For\[\text{9}\text{.89,}\left( \frac{\text{ }\!\!\pm\!\!\text{ 0}\text{.01}}{\text{9}\text{.89}} \right)\text{ }\!\!\times\!\!\text{ 100  }\!\!%\!\!\text{ =}\pm 0.\text{1  }\!\!%\!\!\text{ }\]

Therefore, the relative error will be dependent upon the number itself.

3. The results in the intermediate step of a multi-step computation should be found to have one significant figure more in all the measurements than the number of digits in the least precise measurement.

For example:\[\frac{1}{9.58}=0.1044\]

Now, \[\frac{1}{0.104}=9.56\] and \[\frac{1}{0.1044}=9.58\]

Therefore, taking one extra digit will provide more precise outputs and reduce rounding-off errors.

Dimensions of a Physical Quantity

The powers (exponents) to which base quantities are raised to represent that quantity can be defined as dimensions of a physical quantity. They are figured as the square brackets around the quantity.

• Dimensions of the 7 base quantities have been considered as – Length [L], time [T], Mass [M], thermodynamic temperature [K], luminous intensity [cd], electric current [A] and amount of substance [mol].

For example,

$\text{Volume=Length }\!\!\times\!\!\text{ Breadth }\!\!\times\!\!\text{ Height}$

$text{=}\left[ \text{L} \right]\text{ }\!\!\times\!\!\text{ }\left[ \text{L} \right]\text{ }\!\!\times\!\!\text{ }\left[ \text{L} \right]\text{=}{{\left[ \text{L} \right]}^{\text{3}}}$ 

$\text{Force=Mass }\!\!\times\!\!\text{ Acceleration}$

$\text{=}\frac{\left[ \text{M} \right]\left[ \text{L} \right]}{{{\left[ \text{T} \right]}^{\text{2}}}}\text{=}\left[ \text{M} \right]\left[ \text{L} \right]{{\left[ \text{T} \right]}^{\text{-2}}}$

• The other dimensions for a quantity will be always 0. As an example, in the case of volume only length has 3 dimensions but the mass, time

etc will have 0 dimensions. Zero dimension is shown by superscript 0 like \[\left[ {{\text{M}}^{0}} \right]\].

Dimensions will not affect the magnitude of a quantity Dimensional formula and Dimensional Equation

The expression that represents how and which of the base quantities represent the dimensions of a physical quantity is defined as a Dimensional Formula.

An equation we got after equating a physical quantity with its dimensional formula is a Dimensional Equation.

Dimensional Analysis

• The physical quantities that have similar dimensions only can be added and subtracted. This can be named as the principle of homogeneity of dimensions.

• Dimensions are multipliable and can be cancelled as normal algebraic methods.

• Quantities on both sides should always have identical dimensions, in mathematical equations.

• Arguments of special functions such as trigonometric, logarithmic and ratio of similar physical quantities will be dimensionless.

• Equations will be uncertain to the extent of dimensionless quantities.

As an example, say Distance = Speed x Time. In Dimension terms, 

\[\left[ \text{L} \right]\text{=}\left[ \text{L}{{\text{T}}^{\text{-1}}} \right]\text{ }\!\!\times\!\!\text{ }\left[ \text{T} \right]\]

As the dimensions can be cancelled as we do in algebra, dimension \[\left[ \text{T} \right]\] will get cancelled and the equation will be \[\left[ \text{L} \right]\text{=}\left[ \text{L} \right]\].

Applications of Dimensional Analysis

When we check the Dimensional Consistency of equations

• A dimensionally correct equation should have identical dimensions on both sides of the equation.

• There is no need for a dimensionally correct equation to be a correct equation but a dimensionally incorrect equation will be always incorrect. Dimensional validity can be tested but not calculate the correct relationship between the physical quantities.

Example, \[\text{x=}{{\text{x}}_{\text{0}}}\text{+}{{\text{ }\!\!\nu\!\!\text{ }}_{\text{0}}}\text{t+}\left( \frac{\text{1}}{\text{2}} \right)\text{a}{{\text{t}}^{\text{2}}}\]

Or, Dimensionally, \[\left[ \text{L} \right]\text{=}\left[ \text{L} \right]\text{+}\left[ \text{L}{{\text{T}}^{\text{-1}}} \right]\left[ \text{T} \right]\text{+}\left[ \text{L}{{\text{T}}^{\text{-2}}} \right]\left[ {{\text{T}}^{\text{2}}} \right]\]

Where, \[\text{x}\] be the distance travelled in time t,

\[{{\text{x}}_{0}}\]– starting position,

\[{{\nu }_{0}}\]- initial velocity,

\[\text{a}\]– uniform acceleration.

Dimensions on both sides will be [L] because [T] get cancelled out. Therefore this will be a dimensionally correct equation.

Deducing relation among physical quantities

• For deducing a relation among physical quantities, we must know the dependence of one quantity over others (or independent variables) and assume it as a product type of dependence.

• Dimensionless constants will not be obtainable by the use of this method.

We can take an example, 

\[\text{T=k}{{\text{l}}^{\text{x}}}{{\text{g}}^{\text{y}}}{{\text{m}}^{\text{z}}}\]

Or,

$\left[ {{\text{L}}^{\text{0}}}{{\text{M}}^{\text{0}}}{{\text{T}}^{\text{1}}} \right]\text{=}{{\left[ {{\text{L}}^{\text{1}}} \right]}^{\text{x}}}{{\left[ {{\text{L}}^{\text{1}}}{{\text{T}}^{\text{-2}}} \right]}^{\text{y}}}{{\left[ {{\text{M}}^{\text{1}}} \right]}^{\text{z}}}$ 

$\text{                =}\left[ {{\text{L}}^{\text{x+y}}}{{\text{T}}^{\text{-2y}}}{{\text{M}}^{\text{z}}} \right]$

This means that, \[\text{x+y=0,-2y=1 and z=0}\]. So \[\text{x=}\frac{1}{2}\text{,y=-}\frac{1}{2}\text{ and z=0}\].

Hence the original equation will be reduced to \[\text{T=k}\sqrt{\frac{\text{l}}{\text{g}}}\].

Section–A (1 Mark Questions)

1. What is an error? 

Ans. An error is a mistake of some kind causing an error in your results, so the result is not accurate.

2. Find the number of the significant figures in $11\cdot 118\times 10^{-6}V$.

Ans. The number of significant figures is 5 as 10−6 does not affect this number.

3. A physical quantity P is given by $P=\dfrac{A^{5}B^{\dfrac{1}{2}}}{C^{-4}D^{\dfrac{3}{2}}}$ . Find the quantity which brings in the maximum percentage error in P.

Ans. Quantity C has maximum power. So it brings maximum error in P.

4. What is the difference between accuracy and precision?

Ans. Accuracy is an indication of how close a measurement is to the accepted value.

• An accurate experiment has a low systematic error.

Precision is an indication of the agreement among several measurements.  

• A precise experiment has a low random error.

5. Find the value of a for which $m=\dfrac{1}{\sqrt{3}}$ is a root of the equation $am^{2}+(\sqrt{3}-\sqrt{2})m-1=0$ .

Ans. Put $m=\dfrac{1}{\sqrt{3}}\Rightarrow \dfrac{a}{3}+(\sqrt{3}-\sqrt{2})\dfrac{1}{\sqrt{3}}-1=0$

$\Rightarrow a_+(3-\sqrt{6})-3=0$

So, a = $\sqrt{6}$

Section – B (2 Marks Questions)

6. A force F is given by $F=at+bt^{2}$ , where t is time. What are the dimensions of a and b?

Ans. From the principle of dimensional homogeneity $\left [ F \right ]=\left [ at \right ]\therefore \left [ a \right ]=\left [ \dfrac{F}{t} \right ]=\left [ \dfrac{MLT^{-2}}{T} \right ]=\left [ MLT^{-3} \right ]$

Similarly, $\left [ F \right ]=\left [ bt^{2} \right ]\therefore \left [ b \right ]=\left [ \dfrac{F}{t^{2}} \right ]=\left [ \dfrac{MLT^{-2}}{T^{2}} \right ]=\left [ MLT^{-4} \right ]$ .

7. Let l, r, c, and v represent inductance, resistance, capacitance, and voltage, respectively. Find the dimension of 1/rcv in SI units.  

Ans. Dimension of inductance = [M1L2T−2A−2] = [l]

Dimension of capacitance = [M−1L−2T4A2] = [c]

Dimension of resistance = [M1L2T−3A−2] = [r]

Dimension of voltage = [M1L2T−3A−1] = [v]

Dimension of l/rcv = 

M1L2T−2A−2] / [M−1L2T4A2][M1L2T−3A−2]

[M1L2T−3A−1]
= [ML2T−2A−2]/[ML2T−2A−1]

= [A−1]

8. P represents radiation pressure, c represents speed of light and S represents radiation energy striking unit area per sec. Find the nonzero integers x, y, z such that $P^{x}S^{y}c^{z}$ is dimensionless.

Ans. Try out the given alternatives.

When x = 1, y = −1, z = 1

$P^{x}S^{y}c^{z}=P^{-1}S^{-1}c^{1}=\dfrac{Pc}{S}$

$=\dfrac{\left [ ML^{-1}T^{-2} \right ]\left [ LT^{-1} \right ]}{\left [ \dfrac{ML^{2}T^{-2}}{L^{2}T} \right ]}=\left [ M^{0}L^{0}T^{0} \right ]$

9. The length and breadth of a metal sheet are 3.124 m and 3.002 m respectively. Find the area of this sheet up to the correct significant figure is

Ans. $A=3\cdot 124m\times 3\cdot 002m$

$A=\dfrac{9\cdot 738248}{248m^{2}}$

$=9\cdot 378m^{2}$

10. A certain body weighs 22.42 gm and has a measured volume of 4.7 cc. The possible errors in the measurement of mass and volume are 0.01 gm and 0.1 cc. Then find the maximum error in the density.

Ans. $D=\dfrac{M}{V}\therefore %\dfrac{\Delta D}{D}=%\dfrac{\Delta M}{M}+%\dfrac{\Delta V}{V}=\left ( \dfrac{0\cdot 01}{22\cdot 42}-\dfrac{0\cdot 1}{4\cdot 7} \right )\times 100=2%$

5 Important Topics of Class 11 Physics Chapter 1 Units and Measurements:

Importance of Class 11 Physics Chapter 1 Units and Measurement Revision Notes 

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• Focus on key formulas related to measurement, such as those for calculating density, speed, and other derived quantities. Know how and when to use these formulas.

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