MHT CET · Physics · Gravitation
Assuming that the earth is revolving around the sun in circular orbit of radius \(R\), the angular momentum is directly proportional to \(\mathrm{R}^{\mathrm{n}}\). The value of ' \(n\) ' is
- A 2
- B 1.5
- C 1
- D 0.5
Answer & Solution
Correct Answer
(D) 0.5
Step-by-step Solution
Detailed explanation
We know,
Angular momentum \(L=m v R\)...(i)
For a planet revolving around the sun, \(\frac{\mathrm{mv}^2}{\mathrm{R}}=\frac{\mathrm{GMm}}{\mathrm{R}^2}\)
...(C.P.F = Gravitational force)
\(v^2=\frac{G M}{R}\)
\(\mathrm{v}=\sqrt{\frac{\mathrm{GM}}{\mathrm{R}}}\)...(ii)
Putting equation (ii) into (i)
\(\begin{array}{ll}
\therefore & \mathrm{L}=\mathrm{m} \times \sqrt{\frac{\mathrm{GM}}{\mathrm{R}}} \times \mathrm{R}=\mathrm{m} \sqrt{\mathrm{GMR}} \\
\therefore & \mathrm{~L} \propto \mathrm{R}^{1 / 2} \\
\therefore & \mathrm{n}=0.5
\end{array}\)
Angular momentum \(L=m v R\)...(i)
For a planet revolving around the sun, \(\frac{\mathrm{mv}^2}{\mathrm{R}}=\frac{\mathrm{GMm}}{\mathrm{R}^2}\)
...(C.P.F = Gravitational force)
\(v^2=\frac{G M}{R}\)
\(\mathrm{v}=\sqrt{\frac{\mathrm{GM}}{\mathrm{R}}}\)...(ii)
Putting equation (ii) into (i)
\(\begin{array}{ll}
\therefore & \mathrm{L}=\mathrm{m} \times \sqrt{\frac{\mathrm{GM}}{\mathrm{R}}} \times \mathrm{R}=\mathrm{m} \sqrt{\mathrm{GMR}} \\
\therefore & \mathrm{~L} \propto \mathrm{R}^{1 / 2} \\
\therefore & \mathrm{n}=0.5
\end{array}\)
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