Physics

Frequency increases on filing. So, initial frequency of A is 335 Hz.

f = 340 – 5 = 335 Hz

$Z=\sqrt[]{R^2+{\left(X_L-X_C\right)}^2}$

$Z=\sqrt[]{{\left(120\right)}^2+{\left(10-100\right)}^2}=150\Omega$

$\omega =\frac{1}{\sqrt[]{LC}}=\frac{1}{\sqrt[]{10^{-1}*10^{-4}}}=\sqrt[]{10^5}$

$?\omega =2\pi f$

$\Rightarrow f=\frac{10^3}{2\pi \sqrt[]{10}}=50Hz$

$A=A_0{e}^{-\lambda t_1}$ [Radio active decay law]

$\frac{A}{5}=A_0{e}^{-\lambda \left(t_2-t_1\right)}$

$\Rightarrow ln5=\lambda \left(t_2-t_1\right)$

$\Rightarrow Averagelife=\frac{1}{\lambda }=\frac{\left(t_2-t_1\right)}{\mathcal{l}n5}$

Stress = Y * strain

=> $\frac{T_1}{A}=Y*-\frac{\left({\mathcal{l}}_1-\mathcal{l}\right)}{\mathcal{l}}\left(i\right)$

$\frac{T_2}{A}=Y*\frac{\left({\mathcal{l}}_2-\mathcal{l}\right)}{\mathcal{l}}\left(ii\right)$

=> $\frac{T_1}{T_2}=\frac{{\mathcal{l}}_1-\mathcal{l}}{{\mathcal{l}}_2-\mathcal{l}}$

$\Rightarrow \mathcal{l}=\frac{T_1{\mathcal{l}}_2-T_2{\mathcal{l}}_2}{T_1-T_2}=\frac{T_2{\mathcal{l}}_1-T_1{\mathcal{l}}_2}{T_2-T_1}$            

Lyman Series n_f = 1, ni = 2, 3.

Paschen series n_f = 3, ni = 4, 5, 6 -

$\frac{1}{{\lambda }_1}=R{Z}^2\left(\frac{1}{1^2}-\frac{1}{4^2}\right)$

$\frac{1}{{\lambda }_2}=R{Z}^2\left(\frac{1}{3^2}-\frac{1}{4^2}\right)$

$\frac{1/{\lambda }_1}{1/{\lambda }_2}=\frac{1-\frac{1}{16}}{\frac{1}{9}-\frac{1}{16}}\Rightarrow \frac{{\lambda }_2}{{\lambda }_1}=\frac{\frac{15}{16}}{\frac{7}{9*16}}=\frac{15*9}{7}\Rightarrow \frac{{\lambda }_1}{{\lambda }_2}=\frac{7}{135}$

$l=l_{1}sin\omega t+l_{2}cos\omega t=\sqrt[]{l_1^2+l_2^2}\left[\left(\frac{l_1}{\sqrt[]{l_1^2+l_2^2}}\right)sin\omega t+\frac{l_2}{\sqrt[]{l_1^2+l_2^2}}cos\omega t\right]$

$=\sqrt[]{l_1^2+l_2^2}sin\left(\omega t+\alpha \right)=l_{0}sin(\omega t+\alpha$

$l_{rms}=\frac{l_0}{\sqrt[]{2}}=\frac{\sqrt[]{l_1^2+l_2^2}}{\sqrt[]{2}}$

In elliptical orbit, areal velocity is constant

$y=\alpha x-\beta x^2$

$\Rightarrow \frac{dy}{dx}=\alpha -2\beta x=0$

$\Rightarrow x=\frac{\alpha }{2\beta }$

$y_{max}=\alpha *\frac{\alpha }{2\beta }-\beta *{\left(\frac{\alpha }{2\beta }\right)}^2=\frac{{\alpha }^2}{4\beta }$

Range = $2x=\frac{\alpha }{\beta }=\frac{2u^{2}sin\theta .cos\theta }{g}$

On comparing with

y = x tan θ - $\frac{g{x}^2}{2u^{2}co{s}^2\theta }$

mg – T = ma. (1)

T * R = l . (2)

a = R. (3)

With the help of equations (1), (2) and (3), we get

$a=\frac{mg}{m+\frac{l}{R^2}}$

$v=\sqrt[]{2ah}=\omega R$

$\Rightarrow {\omega }^2=\frac{2mgh}{l+m{R}^2}$