A physics class performs an experiment to determine the winner of a race. An empty can, a solid cylindrical battery and a marble roll, all with the same radius) without slipping down an inclined plane of vertical height H (they are all released at the same height). A box slides without friction on another inclined plane of the same height, H. Which object reaches the bottom of the incline first

a) SPHERE must be the winner

b) BOX is most fastest

Explanation:

For this exercise we must use conservation of energy

starting point. Highest part of the plane

         Em₀ = U = m g h

final point. Lowest part of the plane

          Em_f = K = ½ m v² + ½ I w²

Note that as the objects roll, the kinetic energy of rotation is included. Energy is conserved

           Em₀ = Em_f

           mg h = ½ m v² + ½ I w²

the linear and rotational variables are related

          v = w r

          w = v / r

the moment of inertia of the bodies is tabulated

ring (empty can)      I = m r²

cylinder                    I =  ½ m r²

sphere (marbles)    I = 2/5 m r²

         mgh = ½ m v² + ½ I v²/r² = ½ mv² (+ I / mr²)

         2gh / (1 + I / mr²) = v²             (1)

Let's analyze the value of I / mr2

can            I / mr² = mr² / mr² = 1

cylinder    I / mr² = ½ mr² / mr² = ½

sphere      I / mr² = 2/5 mr² / mr² = 2/5

we substitute in equation 1

               v =

can         v = √(2gh / 2) = √ gh

               v = √gh

cylinder   v=  = √(4gh/3)

                v = 1,155 √gh

sphere     v = = √(10gh/7)

                v = 1.20 √gh

therefore the object with the highest speed is the one that takes less time, consequently the SPHERE must be the cattle.

b) for a box on a frictionless surface, there is no rotational kinetic energy

               mgh = ½ m v²

               v = √2gh

               v = 1.41 √(gh)

When comparing with the latter, this would be the one that arrives first