Question

If the universal constant of gravitation ($G$) were decreasing constantly with time, then a satellite in orbit would still maintain its
(A). Radius
(B). Tangential speed
(C). Angular momentum
(D). Period of revolution

Answer 1

The universal constant of gravitation is the constant of proportionality for the coulomb’s law of attraction where one of the bodies is Earth. A satellite in space revolves around the earth due to the gravitational force which provides it centripetal acceleration. As the value of constant gravitation changes, the force of gravity also changes.

Complete answer:
The centripetal force is given by-
${{F}_{c}}\propto \dfrac{{{v}^{2}}}{r}$ - (1)
Here, ${{F}_{c}}$ is the centripetal force
$v$ is the velocity
$r$ is the radius of the path
The centripetal force acting on the satellite is provided by the gravitational force. From eq (1), as the centripetal force changes due to change in gravitational constant, the ratio of ${{v}^{2}}$ and $r$ will also change and hence their values.
The torque due to centripetal force is always zero. Therefore, the angular momentum of the satellite is conserved.
$\text{mvr = constant}$
Here, $m$ is the mass of the satellite and it is constant
The product of $vr$ will be constant.
The period of revolution of a satellite is given by-
$T=2\pi \sqrt{\dfrac{r}{g}}$
Here, $T$ is the period of revolution
$r$ is the radius of the orbit of satellite
$g$ is the acceleration due to gravity
From the above equation, since radius and acceleration due to gravity are variable because of change in the value of $G$. The period of revolution also varies.
Therefore, the angular momentum of the satellite is constant.

Hence, the correct option is (C).

Note:
The torque acting on a body is the cross product of force and radius vector. In centripetal force, the angle between force and radius vector is zero hence, torque is zero. The centripetal force acts towards the centre. The velocity is the tangential speed whose direction changes at every point.