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 this
amount:

a. sqrt(5/2)
b. sqrt(10/7)
c. sqrt(2)
d. sqrt(14/5)
e. sqrt(6/3)

Answer 1

Answer:

b. √(10/7)

Explanation:

The translational and rotational energy of a wheel at the bottom of the ramp is:

KE + RE = ½ mv² + ½ Iω²

For a hoop, I = mr².

KE + RE = ½ mv² + ½ mr²ω²

KE + RE = ½ mv² + ½ mv²

KE + RE = mv²

For a sphere, I = ⅖ mr².

KE + RE = ½ mv² + ½ (⅖ mr²) ω²

KE + RE = ½ mv² + ⅕ mr²ω²

KE + RE = ½ mv² + ⅕ mv²

KE + RE = ⁷/₁₀ mv²

Since the total energy is the same:

⁷/₁₀ mv₂² = mv₁²

v₂² = ¹⁰/₇ v₁²

v₂ = v₁ √(10/7)