- Home
- Standard 11
- Physics
A particle of mass $m$ is under the influence of the gravitational field of a body of mass $M(\gg m)$. The particle is moving in a circular orbit of radius $r_0$ with time period $T_0$ around the mass $M$. Then, the particle is subjected to an additional central force, corresponding to the potential energy $V_c(r)=m \alpha / r^3$, where $\alpha$ is a positive constant of suitable dimensions and $r$ is the distance from the center of the orbit. If the particle moves in the same circular orbit of radius $r_0$ in the combined gravitational potential due to $M$ and $V_c(r)$, but with a new time period $T_1$, then $\left(T_1^2-T_0^2\right) / T_1^2$ is given by [ $G$ is the gravitational constant.]
$\frac{3 \alpha}{G M r_0^2}$
$\frac{\alpha}{2 G M r_0^2}$
$\frac{\alpha}{G M r_0^2}$
$\frac{2 \alpha}{G M r_0^2}$
Solution
$F _1=\frac{ GMm }{ r _0^2}$
$F _2=\frac{ GMm }{ I _0^2}-\frac{3 m \alpha}{ r _0^4}$
$\frac{\omega_1^2}{\omega_0^2}=\frac{ F _2}{ F _1}=\frac{\frac{ GM }{ r _0^2}-\frac{3 \alpha}{ I _0^4}}{\frac{ GM }{ r _0^2}}$
$\frac{ T _0^2}{ T _1^2}=1-\frac{3 \alpha}{ GMr _0^2}$
$\frac{ T _1^2- T _0^2}{ T _1^2}=\frac{3 \alpha}{ GMr _0^2}$
