By balancing gravitational force and centripetal force, we get \(\dfrac{m v^{2}}{r}=\dfrac{G M m}{r^{2}}\) \(v=\sqrt{\dfrac{G M}{R+h}}(\because r=R+h)\)
\(T=\dfrac{2 \pi(R+h)}{v}=\dfrac{2 \pi(R+h)^{3 / 2}}{\sqrt{G M}}\) \(T^{2}=\dfrac{4 \pi^{2}(R+h)^{3}}{G M}\) \((R+h)^{3}=\dfrac{T^{2} G M}{4 \pi^{2}}\) \(R+h=\left(\dfrac{T^{2} G M}{4 \pi^{2}}\right)^{1 / 3}\) \(\therefore h=\left(\dfrac{T^{2} g R^{2}}{4 \pi^{2}}\right)^{1 / 3}-R\) \(\left(\because G M=g R^{2}\right)\)
JEE - 2024
PHXI08:GRAVITATION
359793
If orbital velocity of a planet is given by \(v=G^{a} M^{b} R^{c}\), then what is the value of \(\frac{{2a + b - 3c}}{{3\;b}}\) ? [where, \(G = \) gravitational constant, \(M = \) mass of planet, \(R = \) Radius of orbit]
359794
Assertion : An astronaut experiences weightlessness in a space satellite. Reason : A body falls freely in space satellite.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
Weightlessness experienced by an astronaut is due to the balance between the centripetal force of gravity and the outward-acting centrifugal force. So correct option is (1).
PHXI08:GRAVITATION
359795
The rotation of the earth having radius \(R\) about its axis speeds up to a value such that a man at latitude angle \(60^{\circ}\) feels weightlessness. The duration of the day in such a case
1 \(\pi \sqrt{\dfrac{R}{g}}\)
2 \(\frac{\pi }{2}\sqrt {\frac{R}{g}} \)
3 \(\frac{\pi }{3}\sqrt {\frac{R}{g}} \)
4 \(\pi \sqrt{\dfrac{g}{R}}\)
Explanation:
\(0=g-R \omega^{2}\) or \({\omega ^2} = \frac{{4\;g}}{R}\) or \({\omega ^2} = 2\sqrt {\frac{g}{R}} \) \(\frac{{2\pi }}{T} = 2\sqrt {\frac{g}{R}} \) \(\therefore T = \pi \sqrt {\frac{g}{R}} \)
By balancing gravitational force and centripetal force, we get \(\dfrac{m v^{2}}{r}=\dfrac{G M m}{r^{2}}\) \(v=\sqrt{\dfrac{G M}{R+h}}(\because r=R+h)\)
\(T=\dfrac{2 \pi(R+h)}{v}=\dfrac{2 \pi(R+h)^{3 / 2}}{\sqrt{G M}}\) \(T^{2}=\dfrac{4 \pi^{2}(R+h)^{3}}{G M}\) \((R+h)^{3}=\dfrac{T^{2} G M}{4 \pi^{2}}\) \(R+h=\left(\dfrac{T^{2} G M}{4 \pi^{2}}\right)^{1 / 3}\) \(\therefore h=\left(\dfrac{T^{2} g R^{2}}{4 \pi^{2}}\right)^{1 / 3}-R\) \(\left(\because G M=g R^{2}\right)\)
JEE - 2024
PHXI08:GRAVITATION
359793
If orbital velocity of a planet is given by \(v=G^{a} M^{b} R^{c}\), then what is the value of \(\frac{{2a + b - 3c}}{{3\;b}}\) ? [where, \(G = \) gravitational constant, \(M = \) mass of planet, \(R = \) Radius of orbit]
359794
Assertion : An astronaut experiences weightlessness in a space satellite. Reason : A body falls freely in space satellite.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
Weightlessness experienced by an astronaut is due to the balance between the centripetal force of gravity and the outward-acting centrifugal force. So correct option is (1).
PHXI08:GRAVITATION
359795
The rotation of the earth having radius \(R\) about its axis speeds up to a value such that a man at latitude angle \(60^{\circ}\) feels weightlessness. The duration of the day in such a case
1 \(\pi \sqrt{\dfrac{R}{g}}\)
2 \(\frac{\pi }{2}\sqrt {\frac{R}{g}} \)
3 \(\frac{\pi }{3}\sqrt {\frac{R}{g}} \)
4 \(\pi \sqrt{\dfrac{g}{R}}\)
Explanation:
\(0=g-R \omega^{2}\) or \({\omega ^2} = \frac{{4\;g}}{R}\) or \({\omega ^2} = 2\sqrt {\frac{g}{R}} \) \(\frac{{2\pi }}{T} = 2\sqrt {\frac{g}{R}} \) \(\therefore T = \pi \sqrt {\frac{g}{R}} \)
By balancing gravitational force and centripetal force, we get \(\dfrac{m v^{2}}{r}=\dfrac{G M m}{r^{2}}\) \(v=\sqrt{\dfrac{G M}{R+h}}(\because r=R+h)\)
\(T=\dfrac{2 \pi(R+h)}{v}=\dfrac{2 \pi(R+h)^{3 / 2}}{\sqrt{G M}}\) \(T^{2}=\dfrac{4 \pi^{2}(R+h)^{3}}{G M}\) \((R+h)^{3}=\dfrac{T^{2} G M}{4 \pi^{2}}\) \(R+h=\left(\dfrac{T^{2} G M}{4 \pi^{2}}\right)^{1 / 3}\) \(\therefore h=\left(\dfrac{T^{2} g R^{2}}{4 \pi^{2}}\right)^{1 / 3}-R\) \(\left(\because G M=g R^{2}\right)\)
JEE - 2024
PHXI08:GRAVITATION
359793
If orbital velocity of a planet is given by \(v=G^{a} M^{b} R^{c}\), then what is the value of \(\frac{{2a + b - 3c}}{{3\;b}}\) ? [where, \(G = \) gravitational constant, \(M = \) mass of planet, \(R = \) Radius of orbit]
359794
Assertion : An astronaut experiences weightlessness in a space satellite. Reason : A body falls freely in space satellite.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
Weightlessness experienced by an astronaut is due to the balance between the centripetal force of gravity and the outward-acting centrifugal force. So correct option is (1).
PHXI08:GRAVITATION
359795
The rotation of the earth having radius \(R\) about its axis speeds up to a value such that a man at latitude angle \(60^{\circ}\) feels weightlessness. The duration of the day in such a case
1 \(\pi \sqrt{\dfrac{R}{g}}\)
2 \(\frac{\pi }{2}\sqrt {\frac{R}{g}} \)
3 \(\frac{\pi }{3}\sqrt {\frac{R}{g}} \)
4 \(\pi \sqrt{\dfrac{g}{R}}\)
Explanation:
\(0=g-R \omega^{2}\) or \({\omega ^2} = \frac{{4\;g}}{R}\) or \({\omega ^2} = 2\sqrt {\frac{g}{R}} \) \(\frac{{2\pi }}{T} = 2\sqrt {\frac{g}{R}} \) \(\therefore T = \pi \sqrt {\frac{g}{R}} \)
By balancing gravitational force and centripetal force, we get \(\dfrac{m v^{2}}{r}=\dfrac{G M m}{r^{2}}\) \(v=\sqrt{\dfrac{G M}{R+h}}(\because r=R+h)\)
\(T=\dfrac{2 \pi(R+h)}{v}=\dfrac{2 \pi(R+h)^{3 / 2}}{\sqrt{G M}}\) \(T^{2}=\dfrac{4 \pi^{2}(R+h)^{3}}{G M}\) \((R+h)^{3}=\dfrac{T^{2} G M}{4 \pi^{2}}\) \(R+h=\left(\dfrac{T^{2} G M}{4 \pi^{2}}\right)^{1 / 3}\) \(\therefore h=\left(\dfrac{T^{2} g R^{2}}{4 \pi^{2}}\right)^{1 / 3}-R\) \(\left(\because G M=g R^{2}\right)\)
JEE - 2024
PHXI08:GRAVITATION
359793
If orbital velocity of a planet is given by \(v=G^{a} M^{b} R^{c}\), then what is the value of \(\frac{{2a + b - 3c}}{{3\;b}}\) ? [where, \(G = \) gravitational constant, \(M = \) mass of planet, \(R = \) Radius of orbit]
359794
Assertion : An astronaut experiences weightlessness in a space satellite. Reason : A body falls freely in space satellite.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
Weightlessness experienced by an astronaut is due to the balance between the centripetal force of gravity and the outward-acting centrifugal force. So correct option is (1).
PHXI08:GRAVITATION
359795
The rotation of the earth having radius \(R\) about its axis speeds up to a value such that a man at latitude angle \(60^{\circ}\) feels weightlessness. The duration of the day in such a case
1 \(\pi \sqrt{\dfrac{R}{g}}\)
2 \(\frac{\pi }{2}\sqrt {\frac{R}{g}} \)
3 \(\frac{\pi }{3}\sqrt {\frac{R}{g}} \)
4 \(\pi \sqrt{\dfrac{g}{R}}\)
Explanation:
\(0=g-R \omega^{2}\) or \({\omega ^2} = \frac{{4\;g}}{R}\) or \({\omega ^2} = 2\sqrt {\frac{g}{R}} \) \(\frac{{2\pi }}{T} = 2\sqrt {\frac{g}{R}} \) \(\therefore T = \pi \sqrt {\frac{g}{R}} \)
