You are building a rube goldberg device. a 3 kg ball rolls down an incline and through a loop with radius 5m. how high does the initial incline have to be for the ball to stay attached to the track at all times?

Given data

mass of the ball = 3 kg ;  

loop radius (r) = 5m,

Determine the height  to stay the ball on an incline track (h) = ?

We know that from loop the loop concept,

                                        h = (5r/2)

                                        Where, r = radius of the loop,

Derivation of equation for h;

           From Newtons II law F = m .a  or  F = m. g

           Net force acting at the top of the loop,

                                 Fr = F + m.a  

                                          since  two forces acting down at the top

                         m. V²/r = m. g          

                                        where  

                                         m. V²/r  = net force  

                                         since a = g(ball rolls due to gravitational force)

                              V²/r = g

                               V = √(g.r)   (i)

The kinetic energy is  acting at the top of the loop which is equal to potential energy (from energy conservation)

                        m.g (2r) + 1/2 m.g.r = m.g.h

                                   2r + 1/2 .r = h

                      Further simplification of above equation

                                   h = 5r/2

                                              end of equation derivation

So, to determine the height of incline, such that ball should  stay attached to the track is given by

                                 h = 5r/2

                                     = (5 × 5)/2

                                     = 12.5 m

            The ball to stay attached to the track, the height of incline should be 12.5 m.