Let's start by looking at a simpler equation: F=ma
Half a: F=ma/2=(1/2)ma; equivalent to halving F
then Double F: F=2(1/2)ma=ma
Or to visualize, let's think of it this way. What is force? It is normally described as an influence that causes a free body to undergo acceleration. If we half that acceleration, while keeping mass constant, the force must be halved (must have been halved is more appropriate here, try to think why this is). But under the same circumstances, if we want to double the force (with acceleration now halved), the only way to do that would be to either double the halved acceleration (not a direct doubling of acceleration, but a doubling of force that is reflected by the doubling of acceleration, again, think why this is), or to double the mass.
Now,
Fc=mv^2/r
Double r: Fc=mv^2/(2r)=(1/2)mv^2/r; equivalent to halving Fc
Continued from above, double Fc: Fc= 2*(1/2)(mv^2/r) = mv^2/r
In order to double Fc, while r is halved, the only way to do so would be to double the halved r, double the mass, or increase the velocity by 2^(1/2).
The term "doubling the force" is a somewhat vague, weird concept, somewhat akin to the chicken and egg idea. Which comes first? A change in force or a change in acceleration? A change in force is reflected by a change in acceleration, but a change in acceleration could have only happened due to a change in force (with mass constant). Think about these questions: Do we change the force by changing the external "influence," or do we change the force by making changes in the body, or the system? How can we tell which one of the two has changed, affecting the value of force? What indicates that the net force has changed?
Of course, all this complexity and potentially maddening recursive thought can be simplified by just looking at the equations and their proportional relationships. This is the MCAT way. 👍
