Deriving some uniform circular motion equations

Question

My question basically boils down to this. How do we derive these relationships.

1.)What is the relationship between radius and centripetal force? (inverse, but why?)

2.)What is the relationship between velocity and centripetal force? ( directly proportional to the square of the velocity, but why?)

3.)What is the relationship between period and centripetal force? (inverse, but why?)

4.)Why does the centripetal force increase if we move an object away from the center of motion?

TL;DR basically what i want to know is how would you derive each of these relationships symbolically.

EDIT: someone asked me to provide some context so i will, we were doing a turntable lab in my physics class, and our teacher asked us to derive these equations from the centripetal force equation.

Answer 1

What is the relationship between radius and centripetal force?

You start from the second law of motion:

$F=ma$

You write the law for each axis.

$F_x=ma_x$, where $a_x = \frac {dv_x}{dt} = \frac {d(v*cos(\omega t))}{dt} = - v *\omega*sin(\omega t) $

$F_y=ma_y$, where $a_y = \frac {dv_y}{dt} = \frac {d(v*sin(\omega t))}{dt} = v *\omega*cos(\omega t)$

You know that $v=2*\pi*R*no. rotations/sec=R\omega$

and you get:

$a=(a_x^2+a_y^2)^\frac {1}{2}=R\omega^2$ and $F=m\omega^2R=m\frac {v^2}{R}$