question_answer

The modon of planets in the solar system is an example of conservation of

A) mass                                     

B) momentum

C) angular momentum       

D) kinetic energy

Correct Answer: C

Solution :

Key Idea: Areal speed of planet is constant. From Kepler's second law of motion, a line joining any planet to the sun sweeps out equal areas in equal intervals of time. Let any instant t, the planet is in position A. Then area swept out be SA is dA = area of the curved triangle SAB \[=\frac{1}{2}(AB\times SA)\] \[=\frac{1}{2}(rd\theta \times r)=\frac{1}{2}{{r}^{2}}d\theta \] The instantaneous areal speed is \[\frac{dA}{dt}=\frac{1}{2}{{r}^{2}}\left( \frac{d\theta }{dt} \right)=\frac{1}{2}{{r}^{2}}\omega \] Let J be angular momentum I the moment of inertia and m the mass, then \[J=I\omega =m{{r}^{2}}\omega \] \[\therefore \]  \[\frac{dA}{dt}=\frac{J}{2m}=\text{constant}\] Hence, angular momentum of the planet is conserved.