Answered By Emily D

You'll need an equation from Kinematics and an equation from Dynamics for this:

 $\left|a_c\right|=\frac{v^2}{r}$|a_c|=v²r   and  $F=m\cdot a$F=m·a 

In this case, we're replacing that r with L for or original and 3L for our tripled rope length

 $a_A=\frac{v^2}{L}$a_A=v²L   and  $a_B=\frac{v^2}{3L}$a_B=v²3L  

Now let's compare what happened between a_A and a_B:

- our velocities did not change

- our denominator (the radius of the curve traveled by the ball) tripled

If our denominator increases, then our acceleration must have decreased! To prove this, let's substitute a_A into our equation for a_B

 $a_A=\frac{v^2}{L}$a_A=v²L   and  $a_B=\frac{1}{3}\cdot\frac{v^2}{L}$a_B=13 ·v²L  

 $a_B=\frac{1}{3}\cdot a_A$a_B=13 ·a_A 

Now we almost have our solution - let's figure out how this affects our inward force!

 $F=m\cdot a$F=m·a 

  $F_A=m\cdot a_A$F_A=m·a_A   and   $F_B=m\cdot\frac{1}{3}\cdot a_A$F_B=m·13 ·a_A  

since the masses have not changed between our two scenarios, the ONLY thing affecting the final force is our changes to the acceleration!

 $F_B=\frac{1}{3}F_A$F_B=13 F_A