Two satellites S and S’ revolve around the earth at distances \(3R\) and \(6R\) from the centre of earth. Their periods of revolution will be in the ratio
Using Kepler's Third Law, \(T^2 \propto r^3 \Rightarrow \frac{T_1}{T_2} = \left(\frac{r_1}{r_2}\right)^{3/2} = \left(\frac{3R}{6R}\right)^{3/2} = \left(\frac{1}{2}\right)^{1.5} = \frac{1}{2^{1.5}}\). Hence, the ratio is 1 : \(2^{1.5}\).
A geostationary satellite has an orbital period of
A geostationary satellite remains stationary relative to the Earth's surface, meaning its orbital period must equal the rotation period of the Earth, which is 24 hours.
For energy of satellite, match the columns (symbols have their respective meaning):
**Column-I**
(i) Kinetic energy
(ii) Potential energy
(iii) Total energy
**Column-II**
(p) \(\frac{L^2}{2mr^2}\)
(q) \(-\frac{L^2}{mr^2}\)
(r) \(-\frac{L^2}{2mr^2}\)
Kinetic energy \(K = \frac{1}{2}mv^2 = \frac{L^2}{2mr^2}\) (since \(L = mvr\)). Potential energy \(U = -\frac{GMm}{r} = -\frac{L^2}{mr^2}\). Total energy \(E = K + U = -\frac{L^2}{2mr^2}\). Thus (i)-p, (ii)-q, (iii)-r.
Two identical satellites are at height \(R\) and \(7R\) from earth surface, the ratio of their kinetic energies will be :
Kinetic energy of a satellite is \(K = \frac{GMm}{2r}\). Here \(r_1 = R + R = 2R\) and \(r_2 = R + 7R = 8R\). The ratio \(frac{K_1}{K_2} = \frac{r_2}{r_1} = \frac{8R}{2R} = 4\).
The energy required to put a satellite of mass \(m\) from earth surface into a orbit of radius \(2R\) is \(E_1\). The energy further needed to change the orbit of this satellite from its present orbit to radius \(4R\) is \(E_2\). The ratio \(\frac{E_1}{E_2}\) is (where \(R\) is radius of earth:
The energy required to put a satellite in orbit from earth's surface is \(E_1 = -\frac{GMm}{2(2R)} - \left(-\frac{GMm}{R}\right) = \frac{3GMm}{4R}\). The energy to change orbit from \(2R\) to \(4R\) is \(E_2 = -\frac{GMm}{2(4R)} - \left(-\frac{GMm}{2(2R)}\right) = \frac{GMm}{8R}\). Thus, \(\frac{E_1}{E_2} = 6\), which gives the ratio \(6 : 1\).
Assertion (A): The mechanical energy of earth-moon system remains same when a heavenly body passes nearby the earth-moon system.
Reason (R): Force exerted by heavenly body on the earth-moon system is non-conservative.
Assertion (A) is false:
An external heavenly body exerts a gravitational force on the Earth-Moon system, causing its mechanical energy to change.
Reason (R) is false: Gravitational force is a conservative force by nature. Therefore, both Assertion (A) and Reason (R) are false.
Assertion (A): Comets move around the sun in elliptical orbits. The gravitational force on the comet due to sun is not normal to the comet’s velocity but the work done by the gravitation force over every complete orbit of the comet is zero.
Reason (R): Gravitational force is a conservative force.
Gravitational force is a conservative force. For a conservative force, the work done over a closed path (like a complete elliptical orbit) is zero.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of the Assertion.
Assertion (A): Two satellites A and B are in the same orbit around the earth, B being behind A. Satellite B can overtake satellite A by increasing its speed.
Reason (R): Orbital speeds of two satellite in same orbit may different
Assertion (A) is false. For a satellite to remain in a given orbit, its speed must be constant. Increasing speed will cause the satellite to move to a higher orbit or escape. Reason (R) is false. Satellites in the same orbit must have the same orbital speed to maintain that orbit. Therefore, both (A) and (R) are false.
Assertion (A): The mechanical energy of earth-moon system remains same when another heavenly body passes nearby the earth-moon system.
Reason (R): Force exerted by heavenly body on the earth-moon system is non-conservative.
Assertion (A) is false.
If another heavenly body passes nearby, it exerts an external gravitational force on the earth-moon system. This external force can do work, changing the system's total mechanical energy.
Reason (R) is false. Gravitational force is a conservative force, not non-conservative. Therefore, both (A) and (R) are false.
Assertion (A): An astronaut in an orbiting space station above the earth experiences weightlessness.
Reason (R): An object orbiting around the earth under the influence of the earth’s gravitational force is in a state of free fall.
Assertion (A) is true.
Astronauts in an orbiting space station experience apparent weightlessness because they, along with the station, are continuously falling towards the Earth.
Reason (R) is true. Orbiting is a continuous state of free fall where the object's tangential velocity prevents it from hitting the Earth. (R) correctly explains (A) because weightlessness is a direct consequence of being in a constant state of free fall.