A15kg child is riding a merry go round at 3rpm. what centripetal force must she exert if she is 8m from the center

This is another application of Newton's second law of motion

         Force = (mass) x (acceleration).

The quantities are a little more complicated in circular motion than
they are in plain old straight-line motion.  In circular motion, the 'force'
is the centripetal force on the object, always pointing toward the center
of the circle.  And the 'acceleration' is the centripetal acceleration, also
pointing toward the center of the circle, and equal to

                        (speed)² / (radius).

The only thing we really need to find is the centripetal acceleration, and
then we'll have everything needed to plug into the formula and calculate
the centripetal force.

     Acceleration =    (speed)² / (radius).

     Speed = (distance) / (time) =

                   (3 circumferences) / (minute) =

                   (3 x 2pi x radius) / minute =

                    (6 pi x 8 meters) / (60 sec) =  0.8 pi m/s .

       Acceleration = (speed)² / (radius) =

                                 0.64 pi²  m²/s²  /  (8 meters) = 0.08 pi²  m/s²  =

                                                                               0.79 m/s² (rounded) .

       Force = (mass) x (acceleration) =

                       (15 kg) x (0.79 m/s²)  =  11.84 newtons
                                                    (about 2.66 pounds) .

I would not say that she "exerts this centripetal force".  The fact of
the matter is much simpler:  This is the force that something must
exert on her, pointing toward the center, in order to keep her revolving
around the center at that speed.  It could be the friction between her
shoes and the platform, if she's standing on the merry-go-round. 
It could be a rope tied between her ankle and something at the center
of the rotating platform.  It could be the safety belt on the horse that
she's riding.  Whatever it is, something has to constantly pull her toward
the center of the platform, with a force of 11.84 newtons, otherwise her
15kg body will not travel in that circular path.

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