Discussion Forum



1)

The potential energy of mass m at a distance r from a fixed point O is  given  by $V(r)=\frac{kr^{2}}{2}$ , where k is a positive constant of appropriate dimensions.  This particle is moving in a circular orbit of radius R  about the point O. If v is the speed of the particle and L is the magnitude of its angular momentum about O, which of the following statements is (are) true?


A) $V= \sqrt{\frac{k}{2m}}R$

B) $V= \sqrt{\frac{k}{m}}R$

C) $L= \sqrt{mk}R^{2}$

D) $L=\sqrt{\frac{mk}{2}}R^{2}$

Answer:

Option B,C

Explanation:

$V=\frac{Kr^{2}}{2}$

$F=-\frac{\text{d}V}{\text{d}r}= -Kr$   [ towars centre]

                                                 [ $F=-\frac{\text{d}V}{\text{d}r}$ ]

 2182019490_cir.JPG

$kR= \frac{mv^{2}}{R}$  [Centripetal force ]

$v=\sqrt{\frac{kR^{2}}{m}}=\sqrt{\frac{k}{m}}R$

  $\Rightarrow$                      $L=mvR=\sqrt{\frac{k}{m}}R^{2}$