1. Mass of a box = 2.3 kg.

Two marbles of mass.

2. 15g and 12.39 g are placed in the box

Total mass of the box correct up to the number of significant digits.

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.

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

$=10.00 \mathrm{m}$

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

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

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

Correct up to 4 significant digits

$=175.1 \mathrm{cm}^2$

$T=2 \pi \sqrt{\frac{7(R-r)}{5 g}}$

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

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

5 experiments:

$T$ 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 $=0.01 \mathrm{s}$

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

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

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

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 $r=\frac{\Delta r}{r} \times 100$

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

$g=\frac{28 \pi^2}{5}\left(\frac{R-r}{T^2}\right)$

$\frac{\Delta g}{g}=\frac{\Delta(R-r)}{R-r}+2 \frac{\Delta T}{T}$

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

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

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

% can be oftained by mulliplying by 100.

$\frac{\Delta g}{g} \times 100$

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

$=4+7.14=11.14$%

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.

Quantities of some dimensions com be added or subtracted.

Cannot add Force and velocity.

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

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

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

$A=B$

Certain quantities which are dimensionless.

Examples

Angles.

Ratios of similar physical quantities.

Refractive Index.

Example

Time period $T$ of vibrator of a drop depends on surface tension $S$, radius and density $\rho$ of liquid.

Find an expression for $T$.

$T \propto s^\alpha \gamma^\beta \rho^\gamma$

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

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

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

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

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

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

$\left[L^0 M^0 T\right]=\left[M T^{-2}\right]^\alpha[L]^\beta\left[m L^{-3}\right]^\gamma$

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 $\sqrt{\frac{r^3 \rho}{S}}$.

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 $A B$.

$[A B]: \rightarrow$ find dimensions of $A$ dimensions of $B$.

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

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

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

${[A] } =\frac{[U]\left[x^2\right]}{[\sqrt{x}]}$

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

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

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

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

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

1.  2 Physical quantities which are not related can have same dimensions.

Example

(a) Moment of a force or Torque $[F][d]$.

(b) Kinetic energy or work done $[F][d]$.

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.

Example

$y =y_0 \sin (\omega t)$

$s =u t+\frac{1}{2} a t^2$

(d) Limited form of $\pi$ theorem of Buckingham.

$f=\frac{d}{L^2} v$

Formula cannot be derived from dimensional analysis.