Consider a pure-rolling race where two objects A and B are simultaneously releas

Question

Consider a pure-rolling race where two objects A and B are simultaneously released from the top of an inclined plane and roll down to a finish line. For each group below, answer which object will win the race or it will be a tie (A or B or Tie). 1. A: 100g solid cylinder with R = 10cm vs B: 100g hollow cylinder with R = 10cm. 2. A: 100g solid cylinder with R = 10cm vs B: 100g solid sphere with R = 10cm. 3. A: 100g solid cylinder with R = 10cm vs B: 200g solid cylinder with R = 10cm. 4. A: 100g solid cylinder with R = 10cm vs B: 100g solid cylinder with R = 20cm.

Explanation / Answer

in case of pure rolling,

a = (m g sin(theta) - f) / a

and f r = I alpha

f = I a / r^2

ma = m g sin(theta) - I (a / r^2)

a = m g sin(theta) / (m + I/r^2)

g sin(theta will be same so we have to compare term X = m / (m + I/r^2))

1. A: X = 100 / (100 + (100 / 2)) = 0.667

B: X = 100 / (100 + 100) = 0.5

A will win.

2. A: X = 0.667

B: X = 100 / (100 + (2 x 100 / 5)) = 0.714

B will win.

3. A: X = 0.667

B: X = 200 / (200 + (200 / 2)) = 0.667

Tie

4. A : X = 0.667

B: X = 100 / (100 + (100/2)) = 0.667

Tie

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