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

length of the ramp, L = 35 m

height, h = L*sin(12) = 7.277 m

a)
here total mechanical energy is conserved

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

g*h = v^2

v = sqrt(g*h) = 8.445 m/s


b)

here total mechanical energy is conserved

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

m*g*h = 0.5*m*v^2 + 0.5*(1/2)*m*r^2*w^2

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

g*h = 0.75*v^2

v = sqrt(g*h/0.75) = 9.75 m/s

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