A particle is moving with a uniform speed in a circular orbit of radius $R $ in a central force inversely proportional to the nth power of $R$. If the period of rotation of the particle is $T$ then -
- $T \propto R^{3/2}$ for any n
- $T \propto R^{\frac{n}{2} + 1}$
- $T \propto R^{(n + 1) / 2 }$
- $T \propto R^{\frac{n}{2} }$
The Correct Option is C
Solution and Explanation
$\Rightarrow \frac{1}{T^{2}} \propto \frac{1}{R^{n+1}}$
$\Rightarrow T \propto R^{\left(\frac{n+1}{2}\right)}$
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Concepts Used:
Gravitation
In mechanics, the universal force of attraction acting between all matter is known as Gravity, also called gravitation, . It is the weakest known force in nature.
Newton’s Law of Gravitation
According to Newton’s law of gravitation, “Every particle in the universe attracts every other particle with a force whose magnitude is,
- F ∝ (M1M2) . . . . (1)
- (F ∝ 1/r2) . . . . (2)
On combining equations (1) and (2) we get,
F ∝ M1M2/r2
F = G × [M1M2]/r2 . . . . (7)
Or, f(r) = GM1M2/r2
The dimension formula of G is [M-1L3T-2].