A wheel is rolling down a ramp. The ramp is 35 m long and inclined at an angle o

Question

A wheel is rolling down a ramp. The ramp is 35 m long and inclined at an angle of 12 degrees from the horizontal. In all cases the wheel starts at rest from the top of the ramp and rolls down the ramp without slipping. (a) If the wheel has a total mass of 1.5 kg, which is distributed in the shape of a thin hoop of radius 24 cm, what is its velocity when it reaches the bottom of the ramp? (b) If the wheel has a total mass of 1.5 kg, which is distributed in the shape of a solid disk of radius 24 cm, what is its velocity when it reaches the bottom of the ramp? (c) If the wheel has a total mass of 1.5 kg, which is distributed in the shape of an inner disk of radius 12 cm and a mass of 0.85 kg, surrounded by a hollow cylinder, also with a mass of 0.65 kg, which has an inner radius of 12 cm and an outer radius of 24 cm, what is its velocity when it reaches the bottom of the ramp? Which of the three wheels do you think would be most appropriate for a car?

Explanation / Answer

L = 35 m, h = L * sin 12 = 7.28 m

a) m g h = 0.5 * m * v2 + 0.5 * I* w2

m g h = 0.5* m * v2 + 0.5 * m * v2

g h = v2

v = sqrt (g h) = sqrt (9.8 * 7.28)

= 8.45 m/s

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b) m g h = 0.5 * m * v2 + 0.5 * I * w2

m g h = 0.5 * m * v2 + 0.5 * (1/2) * m * r2*w2

g h = 0.75 * v2

v = sqrt (g * h / 0.75) = sqrt (9.8 * 7.28 / 0.75)

= 9.75 m/s

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