Physics

Time period of satellite

$\mathrm{T}=2\pi \sqrt[]{\frac{{\mathrm{R}}^3}{\mathrm{G}\mathrm{M}}}$

$=2\pi \sqrt[]{\frac{{\mathrm{R}}^3}{\mathrm{G}\mathrm{d}\frac{4}{3}\pi {\mathrm{R}}^3}}$

$\Rightarrow \mathrm{T}=\sqrt[]{\frac{3\pi }{\mathrm{G}\mathrm{d}}}$ $\Rightarrow \frac{3\pi }{\mathrm{G}\mathrm{d}}={\mathrm{T}}^2$

$\frac{1}{2}\mathrm{m}{\left(\frac{\mathrm{u}}{3}\right)}^2-\frac{1}{2}{\mathrm{m}\mathrm{u}}^2=-{\mathrm{F}}_{\mathrm{R}}*24$

$0-\frac{1}{2}{\mathrm{m}\mathrm{u}}^2=-{\mathrm{F}}_{\mathrm{R}}*\mathrm{d}$

$\frac{\frac{1}{2}{\mathrm{m}\mathrm{u}}^2}{\frac{1}{2}m{u}^2*\frac{8}{9}}=\frac{\mathrm{d}}{24}$

$\mathrm{d}=24*\frac{9}{8}=27\mathrm{c}\mathrm{m}$

$\begin{array}{rr}& {}_{11}^{22}\mathrm{N}\mathrm{a}\to \mathrm{X}+{\mathrm{e}}^{+}+\mathrm{v}\\ & {}_{11}^{22}\mathrm{N}\mathrm{a}\to {}_{10}^{22}\mathrm{N}\mathrm{e}+{\mathrm{e}}^{+}+\mathrm{v}\end{array}$

In the frame of car

$\mathrm{N}=\mathrm{m}\mathrm{g}$

and  $\mathrm{f}=\mathrm{m}\mathrm{a}$

$\mathrm{f}\le \mu \mathrm{N}$

$\Rightarrow \mathrm{a}\le \mu \mathrm{g}$

$\Rightarrow \mathrm{a}\le 1.5{\mathrm{m}\mathrm{s}}^{-2}$

${\mathrm{a}}_{\mathrm{m}\mathrm{a}\mathrm{x}}=1.5{\mathrm{m}\mathrm{s}}^{-2}$

Let height of bridge $=\mathrm{h}$

Displacement of ball, $S=-h$

$\begin{array}{rr}& S=ut+\frac{1}{2}{\mathrm{a}\mathrm{t}}^2\\ & -\mathrm{h}=4*4+\frac{1}{2}(-10)(4{)}^2\\ & \Rightarrow \mathrm{h}=64\mathrm{m}\end{array}$

Factual (theory based)

Capacitor in parallel removes the AC ripple from the rectified output.

$\frac{\mathrm{N}}{{\mathrm{N}}_0}=\frac{1}{16}={\left(\frac{1}{2}\right)}^{\frac{\mathrm{t}}{{\mathrm{T}}_{1/2}}}$

$\mathrm{t}=4{\mathrm{T}}_{1/2}=20*4=80\text{minutes}$

$?\frac{1}{\lambda }=\mathrm{R}\left(\frac{1}{{\mathrm{n}}_2^2}-\frac{1}{{\mathrm{n}}_1^2}\right)$

$\Rightarrow \frac{1}{\lambda }=\mathrm{R}\left(\frac{1}{2^2}\right)$

$\Rightarrow \lambda =\frac{4}{\mathrm{R}}$

$\frac{1}{{\lambda }^{\mathrm{\text{'}}}}=\mathrm{R}\left(\frac{1}{4^2}\right)$

$\Rightarrow {\lambda }^{\mathrm{\text{'}}}=\frac{16}{\mathrm{R}}$

$\Rightarrow {\lambda }^{\mathrm{\text{'}}}=4\lambda$

Speed of light in air is

$\mathrm{C}=\frac{\mathrm{x}}{{\mathrm{t}}_1}$

Speed of light in another denser medium.

${\mathrm{C}}_2=\frac{10\mathrm{x}}{{\mathrm{t}}_2}$

$\Rightarrow \mu =\frac{C}{C_2}=\frac{x}{t_1}*\frac{t_2}{10x}$

$\Rightarrow \mu =\frac{{\mathrm{t}}_2}{10{\mathrm{t}}_1}$

(d) $\mathrm{\Delta }\theta =\frac{\lambda }{\mathrm{d}}$

$\mathrm{\Delta }\theta$ is proportional to $\lambda$ but independent of $D$ .

(c) $\mathrm{e}\mathrm{V}=$ Energy of electron

for minimum wavelength, maximum loss of energy

$\mathrm{e}\mathrm{V}=\left(\frac{\mathrm{h}\mathrm{c}}{\lambda }\right)$

$\lambda \propto \left(\frac{1}{\mathrm{V}}\right)$

31. (b) Incident energy $=2.20\mathrm{e}\mathrm{V}$

If $?<2.20\mathrm{e}\mathrm{V}$  electron will emit.

$?>2.20\mathrm{e}\mathrm{V}\mathrm{}$ No electron emission

Only caesium will emit electron