Question

Your boss wants you to advise him about what shape of wheels to use for his son’s pinewood
derby. This is where a car is released from rest and rolls down the ramp to the finish line under
the influence of gravity. He was thinking of using heavy tungsten wheels with most of the weight
in a hoop shape. Explain to him that a solid sphere will go faster at the bottom than a hoop by what amount?

Answer 1

Answer:

Ring   v² = gh

solid wheel (cylinder)   v² = 4/3 gh

Explanation:

Let's use conservation of energy to find the speed of the wheels at the bottom of the hill.

starting point. Point before starting movement

           Em₀ = mgh

final point. At the bottom of the hill

           Em_f = K = ½ m v² + ½ I w²

energy is conserved

            Emo = Em_f

           mgh = ½ m v² + ½ I w²

angular and linear velocity are related

           v = w r

we substitute

          mgh = ½ m v² + ½ I v² / r²

          mgh = $\frac{1}{2} \ ( m+ \frac{I}{r^2} ) \ v^2$½ (m + I / r²) v²

          v² = $\frac{2mgh}{m+ \frac{I}{r^2} }$

the moments of inertia are tabulated

Ring

      I = mr²

         v² = 2 m g h / (m + m)

         v² = gh

solid wheel (cylinder)

      I = ½ m r²

        v² = 2m gh / (m + m / 2)

        v² = 4/3 gh

We can see that due to the difference in the moment of inertia of each body it is different, the solid wheel has more speed when it reaches the lower part of the ramp