Question

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?

Answer 1

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.