physics-class-11-unit-02-chapter-02-dimensions-of-physical-quantities-dimensional-analysis-l-2-4-85pbnln-0f4

### <Success style="font-size:25px"> Dimensions Of Physical Quantities Dimensional Analysis L-2 </Success>
### <primary style="font-size:24px"> Significant Digits </primary>
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- Total mass of box + marbles.

- 2.3 kg + 0.00215kg + 0.01239 kg.

- =   2.31454 kg.

- =   2.3 kg.

- Mass of box = 2.300 kg.

- Up to 3 decimal: 2.315kg.

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<footer style = "font-size: 18px">Dimensions of Physical Quantities, Dimensional Analysis $\rightarrow$ Significant Digits $\rightarrow$ <span style = "color:blue"> Significant Digits </span> $\rightarrow$ Significant Digits $\rightarrow$ Significant Digits </footer>

### <Success style="font-size:25px"> Dimensions Of Physical Quantities Dimensional Analysis L-2 </Success>
### <primary style="font-size:24px"> Significant Digits </primary>
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- **Two lengths $L_1$ and $L_2$ are measured.**

- $ L_1=9.99 \mathrm{m},$

- $L_2=9.9 \mathrm{mm}$

- Sum   =   ?

- $L_1=9.99 \mathrm{m}, \quad L_2=0.0099 \mathrm{m}$

- $L_1+L_2=9.9999 \mathrm{m}$

- Correct up to 2 decimal place

- $=10.00 \mathrm{m}$

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<footer style = "font-size: 18px">Significant Digits $\rightarrow$ Significant Digits $\rightarrow$ <span style = "color:blue"> Significant Digits </span> $\rightarrow$ Significant Digits $\rightarrow$ Significant Digits </footer>

### <Success style="font-size:25px"> Dimensions Of Physical Quantities Dimensional Analysis L-2 </Success>
### <primary style="font-size:24px"> Significant Digits </primary>
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- **Each side of a cube:**

- $=5402 \mathrm{cm}$

- Find surface area of cube in appropriate significant figures.

- Surface Area =$6 a^2$

- Number of significant digits = 4

- $6 a^2=175.089624 \mathrm{cm}^2$.

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### <Success style="font-size:25px"> Dimensions Of Physical Quantities Dimensional Analysis L-2 </Success>
### <primary style="font-size:24px"> To Determine g use Formula </primary>
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- $T=2 \pi \sqrt{\frac{7(R-r)}{5 g}}$

- $R \rightarrow 60 \pm 1 \mathrm{mm}$

- $r \rightarrow 10 \pm 1 \mathrm{mm}$

- <span style="color:blue">5 experiments:</span>

- $T$ measured as $0.52 \mathrm{s}, 0.56 \mathrm{s}, 0.57 \mathrm{s}$ $0.54 \mathrm{s}$ and $0.59 \mathrm{s}$

- Least count of watch $=0.01 \mathrm{s}$

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<footer style = "font-size: 18px">Significant Digits $\rightarrow$ Significant Digits $\rightarrow$ <span style = "color:blue"> To Determine g use Formula </span> $\rightarrow$ Percentage Error $\rightarrow$ Percentage Error </footer>

### <Success style="font-size:25px"> Dimensions Of Physical Quantities Dimensional Analysis L-2 </Success>
### <primary style="font-size:24px"> Percentage Error </primary>
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- Find % error in measurement of r, T and g.

- $T_{\text {mean }}  =\frac{0.52+0.56+0.57+0.54+0.59}{5} $

- $=0.556=0.56 \mathrm{s}$

- Find $\left|\Delta T_{\text {mean }}\right|$

- $\left|\Delta T_1\right| = \left|0.52-0.56\right| = 0.04$

- $\left|\Delta T_2\right| = |0.59-0.56|=0.03$.

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<footer style = "font-size: 18px">Significant Digits $\rightarrow$ To Determine g use Formula $\rightarrow$ <span style = "color:blue"> Percentage Error </span> $\rightarrow$ Percentage Error $\rightarrow$ Percentage Error </footer>

### <Success style="font-size:25px"> Dimensions Of Physical Quantities Dimensional Analysis L-2 </Success>
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- Mean error $=\frac{0.04+0.00+0.01+0.02+0.03}{5}$

- $=\frac{0.1}{5}=0.02 \mathrm{s}$

- % error in $T=\frac{\Delta T}{T} \times 100$

- $=\frac{0.02}{0.56} \times 100=3.57 \$%

- % error in $r=\frac{\Delta r}{r} \times 100$

- $=\frac{1}{10} \times 100=10$%.

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<footer style = "font-size: 18px">To Determine g use Formula $\rightarrow$ Percentage Error $\rightarrow$ <span style = "color:blue"> Percentage Error </span> $\rightarrow$ Percentage Error $\rightarrow$ Percentage Error </footer>

### <Success style="font-size:25px"> Dimensions Of Physical Quantities Dimensional Analysis L-2 </Success>
### <primary style="font-size:24px"> Percentage Error </primary>
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- $g=\frac{28 \pi^2}{5}\left(\frac{R-r}{T^2}\right) $
 
- $ \frac{\Delta g}{g}=\frac{\Delta(R-r)}{R-r}+2 \frac{\Delta T}{T} $

- $ \Delta(R-r)=\Delta R+\Delta r=1+1=2 {mm} $

- $ (R-r)=60-10=50 {mm} $

- $ \frac{\Delta g}{g}=\frac{2}{50}+\frac{2 \times 0.02}{0.56}$

- % can be oftained by mulliplying by 100.

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<footer style = "font-size: 18px">Percentage Error $\rightarrow$ Percentage Error $\rightarrow$ <span style = "color:blue"> Percentage Error </span> $\rightarrow$ Percentage Error $\rightarrow$ Basic Dimensions </footer>

### <Success style="font-size:25px"> Dimensions Of Physical Quantities Dimensional Analysis L-2 </Success>
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- $ \frac{\Delta g}{g} \times 100 $

- $ =\frac{200}{50}+\frac{2 \times 0.02}{0.56} \times 100 $

- $  =4+7.14=11.14 $%

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<footer style = "font-size: 18px">Percentage Error $\rightarrow$ Percentage Error $\rightarrow$ <span style = "color:blue"> Percentage Error </span> $\rightarrow$ Basic Dimensions $\rightarrow$ Homogeneity Principle of Dimensional Analysis </footer>

### <Success style="font-size:25px"> Dimensions Of Physical Quantities Dimensional Analysis L-2 </Success>
### <primary style="font-size:24px"> Basic Dimensions </primary>
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- **All quantities con be expressed in 7 basic dimensions.**

- [L] $\rightarrow$ Length.

- $[\mathrm{m}] \rightarrow$ mass.

- $[T] \rightarrow$ Time.

- $[A] \rightarrow$ current.

- $[K] \rightarrow$ Temperature.

- $[Cd] \rightarrow$ Luminous Intensity.

- $[mol] \rightarrow$ Moles.

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<footer style = "font-size: 18px">Percentage Error $\rightarrow$ Percentage Error $\rightarrow$ <span style = "color:blue"> Basic Dimensions </span> $\rightarrow$ Homogeneity Principle of Dimensional Analysis $\rightarrow$ Homogeneity Principle of Dimensional Analysis </footer>

### <Success style="font-size:25px"> Dimensions Of Physical Quantities Dimensional Analysis L-2 </Success>
### <primary style="font-size:24px"> Homogeneity Principle of Dimensional Analysis </primary>
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- Quantities of some dimensions com be added or subtracted.

- Cannot add Force and velocity.

- $\text { Speed }=\frac{\text { Dist }}{\text { Time }}\left[\frac{L}{T}\right]=\left[L T^{-1}\right]$

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<footer style = "font-size: 18px">Percentage Error $\rightarrow$ Basic Dimensions $\rightarrow$ <span style = "color:blue"> Homogeneity Principle of Dimensional Analysis </span> $\rightarrow$ Homogeneity Principle of Dimensional Analysis $\rightarrow$ Dimensionless Quantities </footer>

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

- Principle of Dimensional Horrogeniety all terms in same equation which are added or subtracted have same dimensions.

- $A=B$

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<footer style = "font-size: 18px">Basic Dimensions $\rightarrow$ Homogeneity Principle of Dimensional Analysis $\rightarrow$ <span style = "color:blue"> Homogeneity Principle of Dimensional Analysis </span> $\rightarrow$ Dimensionless Quantities $\rightarrow$ Dimensional Analysis </footer>

### <Success style="font-size:25px"> Dimensions Of Physical Quantities Dimensional Analysis L-2 </Success>
### <primary style="font-size:24px"> Dimensionless Quantities </primary>
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- **Certain quantities which are dimensionless.**

- <b>Examples</b>

- Angles.

- Ratios of similar physical quantities.

- Refractive Index.

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<footer style = "font-size: 18px">Homogeneity Principle of Dimensional Analysis $\rightarrow$ Homogeneity Principle of Dimensional Analysis $\rightarrow$ <span style = "color:blue"> Dimensionless Quantities </span> $\rightarrow$ Dimensional Analysis $\rightarrow$ Dimensional Analysis </footer>

### <Success style="font-size:25px"> Dimensions Of Physical Quantities Dimensional Analysis L-2 </Success>
### <primary style="font-size:24px"> Dimensional Analysis </primary>
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- $T \propto s^\alpha \gamma^\beta \rho^\gamma$

- $\alpha, \beta, \gamma$ are unknowns

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

- $ [S]=\frac{\text { Force }}{\text { Length }}$

- $=\left[M \frac{L}{T^2} \cdot \frac{1}{L}\right]=[L^{0} M T^{-2}]$

- ${[r]=\left[L M^0 T^0\right]=[L]}$

- ${[\rho]=\frac{\text { mass }}{\text { Volume }}=\frac{m}{L^3}=\left[M L^{-3}\right]}$.

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<footer style = "font-size: 18px">Dimensionless Quantities $\rightarrow$ Dimensional Analysis $\rightarrow$ <span style = "color:blue"> Dimensional Analysis </span> $\rightarrow$ Dimensional Analysis $\rightarrow$ Dimensional Analysis </footer>

### <Success style="font-size:25px"> Dimensions Of Physical Quantities Dimensional Analysis L-2 </Success>
### <primary style="font-size:24px"> Dimensional Analysis </primary>
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- $\left[L^0 M^0 T\right]=\left[M T^{-2}\right]^\alpha[L]^\beta\left[m L^{-3}\right]^\gamma$

- Equate powers of $L, M, T$ separately.

- L:   $0=\beta-3 \gamma$

- M:   $0=\alpha+\gamma$

- T:   $ 1=-2 \alpha $

- $\alpha=-\frac{1}{2}$,  $\gamma=\frac{1}{2},$   $\beta  =\frac{3}{2}$

- $ T \propto S^\alpha r^\beta \rho^\gamma $

- $T=k S^\frac{-1}{2} r^\frac{3}{2} \rho^{\frac{1}{2}} $ = k $\sqrt{\frac{r^3 \rho}{S}}$.

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<footer style = "font-size: 18px">Dimensional Analysis $\rightarrow$ Dimensional Analysis $\rightarrow$ <span style = "color:blue"> Dimensional Analysis </span> $\rightarrow$ Dimensional Analysis $\rightarrow$ Dimensional Analysis </footer>

### <Success style="font-size:25px"> Dimensions Of Physical Quantities Dimensional Analysis L-2 </Success>
### <primary style="font-size:24px"> Dimensional Analysis </primary>
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- **Example**

-  Potential energy of a particle varies with distance $x$ from origin as:
 
 - $U=\frac{A \sqrt{x}}{x^2+B}$
 
- $A$ and $B$ are dimensional constants. Find the dimensional formula for $A B$.

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### <Success style="font-size:25px"> Dimensions Of Physical Quantities Dimensional Analysis L-2 </Success>
### <primary style="font-size:24px"> Dimensional Analysis </primary>
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- $[A B]: \rightarrow$ find dimensions of $A$ dimensions of $B$.

- $x^2+B \Rightarrow$ dimension of $B$ $=$ dimension of $x^2$.

- $ {[B]=\left[L^2\right]}$.

- $ A=\frac{U\left(x^2+B\right)}{\sqrt{x}}$.

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### <Success style="font-size:25px"> Dimensions Of Physical Quantities Dimensional Analysis L-2 </Success>
### <primary style="font-size:24px"> Dimensional Analysis </primary>
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- ${[A] } =\frac{[U]\left[x^2\right]}{[\sqrt{x}]}$

- ${[U]=} \text { Energy } \rightarrow\left[m v^2\right] =\left[m L^2 T^{-2}\right]$

- ${[A] } =\left[m L^2 T^{-2}\right]\left[L^2\right]\left[L^{-1 / 2}\right]$

- ${[A] } =\left[M L^{7 / 2} T^{-2}\right]$

- ${[A B] }  =\left[m L^{7 / 2} T^{-2}\right]\left[L^2\right]$

- $=\left[m L^{11 / 2} T^{-2}\right]$.

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### <Success style="font-size:25px"> Dimensions Of Physical Quantities Dimensional Analysis L-2 </Success>
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- **2.** Formulas correct up to a consent K. - dimensionless.

- Dimensional analysis cannot help in finding the value of $K$.

- **3.** Cannot use dimensional analysis to predict behaviour where equation have quantities other them powers of products.

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<footer style = "font-size: 18px">Dimensional Analysis $\rightarrow$ Limitations $\rightarrow$ <span style = "color:blue"> Limitations </span> $\rightarrow$ Limitations $\rightarrow$ Frequency of Tuning Fork </footer>