A bicycle and rider roll from rest down an incline with a vertical height of h.

Question

A bicycle and rider roll from rest down an incline with a vertical height of h. The only rotating parts are the two wheels, which have a combined mass of 3.0 kg and can be modeled as hoops with a rotational inertia of MR^2, The total mass of rider and bicycle (including wheels) is 85.0 kg. Use the conservation of energy to derive an equation for the TRANSLATIONAL VELOCITY of the bicycle and rider as a function of height. Your answer should be in the form v = squareroot Cgh where C is a numerical value (unitless). You may solve for C as a decimal value or as a fraction.

Explanation / Answer

For rolling without slipping,

v = w r => w = v / r

Applying energy conservation,

PEi + KEi = PEf + KEf

M g h + 0 = 0 + ( M v^2 /2 + I w^2 / 2 )

and I w^2 = ( m r^2)(v/r)^2 = m v^2

M g h = M v^2 /2 + m v^2 /2

(85 x g x h) = (85 v^2 /2 ) + (6 v^2 / 2)

45.5 v^2 = 85 g h

v = sqrt(1.868 g h )

Ans: C = 1.868

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