Question

A satellite in a force-free space sweeps stationary interplanetary dust at a rate $\dfrac{dM}{dt}=\beta v$. The acceleration of a satellite is?
$\begin{aligned} & A.-\dfrac{\beta {{v}^{2}}}{M} \\\ & B.-\dfrac{\beta {{v}^{2}}}{2M} \\\ & C.-\beta {{v}^{2}} \\\ & D.-\dfrac{M\beta }{{{v}^{2}}} \\\ \end{aligned}$

Answer 1

As the interplanetary dust is swept therefore there is change in mass. So force acting on the satellite is not just mass into acceleration. It is differentiation of momentum with respect to time. As it is given to us that the satellite is force-free, we can equate the force equation to zero and find acceleration.

Formula used:
$F=\dfrac{d}{dt}(Mv)$

Complete step by step answer:
The force acting on the satellite is
$\begin{aligned} & F=\dfrac{d}{dt}(Mv) \\\ & =\dfrac{dv}{dt}M+v\dfrac{dM}{dt} \\\ & =M\dfrac{dv}{dt}+v(\beta v) \\\ \end{aligned}$
We know that the net force is zero.
$F=0$
$\begin{aligned} & M\dfrac{dv}{dt}=-v(\beta v) \\\ & \dfrac{dv}{dt}=a=-\dfrac{\beta {{v}^{2}}}{M} \\\ \end{aligned}$
So, the correct answer is “Option A”.

Additional information:
In Newtonian mechanics, the centrifugal force is an inertial force that appears to act on all objects when placed on a rotating object. The force is directed away from the axis of rotation. The centrifugal force is directed radially outwards if the axis of rotation passes through the origin. The magnitude of centrifugal force F with angular velocity ω on an object of mass m at the distance r from the origin is:
$F=m{{w}^{2}}r$
The concept of centrifugal force has been applied across various fields like sciences, railways, aircrafts, centrifuges for chemical synthesis and planetary orbits. The term has also been used for the reactive centrifugal force that may be acknowledged as a reaction to a centripetal force in some areas.

Note:
The conception of centrifugal force has evolved since the early 1500s by prolific scientists and thinkers who proposed concepts of it at that time. It was debated that centrifugal force provided ideas about absolute motion which is of great importance in classical mechanics. Units of centrifugal force are Newton and dyne.