Auniform ball of radius r rolls without slipping between two rails such that the horizontal distance is d between the two contact points of the rails to the ball. (a) show that at any instant vcm = ω − d2/4. discuss this expression in the limits d = 0 and d = 2r. (b) for a uniform ball starting from rest and descending a vertical distance h while rolling without slipping down a ramp, vcm = /7. replacing the ramp with the two rails, show that v = 10gh . in each case, the cm 5+2/(1−d2 /4r2 ) work done by friction has been ignored. (c) which speed in part (b) is smaller? why? answer in terms of how the loss of potential energy is shared between the gain in translational and rotational kinetic energies. (d) for which value of the ratio d/r do the two expressions for the speed in part (b) differ by 5.0%? by 0.50%?

potential energy is directly proportional to height.

more height = more potential energy.

greatest increase in height = greatest increase in potential energy

where are the points with the greatest increase in height ?

has to be c-e or d-e .

speed of light in vacuum is 300,000 km/s

speed of light in other mediums is given by the expression v = c/n, where c is the speed of light in vacuum and n is the refractive index of medium.

for the first medium v = 200,000 km/s

  so, refractive index of medium = 300000/200000 = 1.5

    1.5 refractive index is usually for glass. so first medium is glass.

for the second medium v = 250,00 km/s

so, refractive index of medium = 300000/250000 = 1.2

    1.2 refractive index is usually for water. so second medium is water.

for the third medium v = 0 km/s

so, refractive index of medium = 300000/0 = ∞

    ∞ refractive index is usually for opaque mediums. so third medium is opaque medium.