Respuesta :
Answer:
The angular frequency of oscillation of the mass is 11 rad/s.
Explanation:
Given that,
Charge = 2μC
Radius R₁= 8 cm
Radius R₂ = 16 cm
Angular frequency = 22 rad/s
We need to calculate the angular frequency of oscillation of the mass
The electric field produced along x axis
[tex]E=\dfrac{kqx}{\sqrt{R^2+x^2}}[/tex]
[tex]E=\dfrac{kqx}{\sqrt{R^2(1+\dfrac{x^2}{R^2})^2}}[/tex]
[tex]E=\dfrac{kqx}{R^3}[/tex]
The force on the mass is
[tex]F=Eq[/tex]
[tex]F=\dfrac{kQqx}{R^3}[/tex]....(I)
For,x<<R
Now, using centripetal force
[tex]F = \dfrac{mv^2}{r}[/tex]
Put the value of F in equation (I)
[tex]\dfrac{mv^2}{r}=\dfrac{kQq}{R^3}[/tex]
We know that,
[tex]v=r\omega[/tex]
[tex]m\omega^2r=\dfrac{kQq}{R^3}[/tex]
[tex]\omega^2=\dfrac{kQq}{mrR^3}[/tex]
For, r<<R
[tex]\omega^2=\dfrac{kQq}{mR^3}[/tex]
[tex]\omega=\sqrt{\dfrac{kQq}{mR^3}}[/tex]
Here,
[tex]\omega\propto\sqrt{\dfrac{q}{R^3}}[/tex]
The ratio of angular frequency
[tex]\dfrac{\omega}{\omega_{1}}=\sqrt{\dfrac{\dfrac{q}{R^3}}{\dfrac{q_{1}}{R_{1}^3}}}[/tex]
[tex]\dfrac{\omega}{\omega'}=\sqrt{\dfrac{R^3\times2q}{2R^3\times q}}[/tex]
[tex]\omega=\sqrt{\dfrac{8^3\times2\times2\times10^{-6}}{8\times8^3\times2\times10^{-6}}}\times\omega'[/tex]
[tex]\omega=0.5\omega'[/tex]
Put the value of
[tex]\omega=\dfrac{1}{2}\times22[/tex]
[tex]\omega=11\ rad/s[/tex]
Hence, The angular frequency of oscillation of the mass is 11 rad/s.
