Question

A hockey puck, a tire and a baseball are rolled down an incline. They all start from rest at the same time and roll smoothly. Which one reaches the bottom first? Which one reaches last? Does the result depend on the size of the objects or their mass? Explain your answers.

Answer 1

let H is the initial height

for puck (disk),

Apply conservation of enrgy

m*g*H = 0.5*m*v^2 + 0.5*I*w^2

m*g*H = 0.5*m*v^2 + 0.5*0.5*m*R^2*w^2

m*g*H = 0.5*m*v^2 + 0.25*m*v^2

m*g*H = 0.75*m*v^2

v_puck = sqrt(g*H/0.75)

= 1.154*sqrt(g*H)

for tire,

Apply conservation of enrgy

m*g*H = 0.5*m*v^2 + 0.5*I*w^2

m*g*H = 0.5*m*v^2 + 0.5*m*R^2*w^2

m*g*H = 0.5*m*v^2 + 0.5*m*v^2

m*g*H = m*v^2

v^2 = g*H

v_tire= sqrt(g*H)

for baseball,

Apply conservation of enrgy

m*g*H = 0.5*m*v^2 + 0.5*I*w^2

m*g*H = 0.5*m*v^2 + 0.5*(2/5)*m*R^2*w^2

m*g*H = 0.5*m*v^2 + 0.2*m*v^2

m*g*H = 0.7*m*v^2

v_baseball = sqrt(g*H/0.7)

= 1.195*sqrt(g*H)

clearly, V_baseball > V_puck > V_tire

so, base ball reaches the bottom first and puck reaches the lost.

The rsult does not depend on size and mass of the objects.