A planet of mass \(m\) is moving in an elliptical orbit about the sun with time period \(T\). If \(A\) be the area of orbit, then its angular momentum would be :
According to Kepler's second law, the areal velocity of the planet is constant and is given by \(\frac{dA}{dt} = \frac{L}{2m}\). Integrating over one time period \(T\) gives \(A = \frac{L}{2m} T ⇒ L = \frac{2mA}{T}\).
Assertion (A): Period of revolution of satellite in circular orbit around earth is inversely proportional to cube of its orbital speed.
Reason (R): Period of revolution in uniform circular motion is given by \( T = \frac{2\pi r}{v} \) where \( r \) is radius of orbit and \( v \) is speed.
Assertion (A) is true. For a satellite in circular orbit, orbital speed \( v = \sqrt{\frac{GM}{r}} \) implying \( r \propto \frac{1}{v^2} \). The period is \( T = \frac{2\pi r}{v} \). Substituting \( r \), we get \( T \propto \frac{1/v^2}{v} \propto \frac{1}{v^3} \).
Reason (R) is true. The formula \( T = \frac{2\pi r}{v} \) is the correct definition for the period of uniform circular motion. However, (R) is a kinematic definition and does not explain the dynamic relationship between \( T \) and \( v \) for a satellite, which requires considering gravity. Thus, (R) is not the correct explanation of (A).
Assertion (A): When planet moves in elliptical orbit around Sun. Its angular momentum about sun remains conserved.
Reason (R): Total mechanical energy of planet – sun system remains conserved.
Assertion (A) is true. Gravitational force is a central force, so the torque about the Sun is zero, leading to angular momentum conservation. Reason (R) is also true. Gravitational force is conservative, so total mechanical energy of the system is conserved. However, energy conservation does not explain angular momentum conservation, as they are distinct conservation laws.
Assertion (A): The radius vector from the sun to a planet sweeps out equal areas in equal times interval.
Reason (R): Transverse (perpendicular to radius vector) acceleration of the planet is zero.
Assertion (A): This statement is Kepler's Second Law, which is a direct consequence of angular momentum conservation. So (A) is true.nReason (R): For a central force, like gravity, the force acts along the radius vector, meaning no transverse force component exists. Thus, transverse acceleration is zero. So (R) is true.n(R) explains (A) because zero transverse acceleration leads to conservation of angular momentum, which implies Kepler's Second Law.
Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): Angular momentum conservation can be used to explain Kepler’s second law of planetary motion.
Reason (R): Areal velocity of a planet revolving around the sun is equal to its angular momentum.
In the light of the above statements, choose the correct answer from the options given below:
Assertion (A) is true because Kepler's second law represents the conservation of angular momentum. Reason (R) is false because areal velocity is \(\frac{dA}{dt} = \frac{L}{2m}\), which is proportional to angular momentum \(L\) but not equal to it.