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%?

Learning goal: to understand newton's 1st law. newton's principia states this first law of motion: an object subject to no net force maintains its state of motion, either at rest or at constant speed in a right line. this law may be restated as follows: if the sum of all forces acting on an object is zero, then the acceleration of that object is zero. mathematically this is just a special case of the 2nd law of motion, f⃗ =ma⃗ , when f⃗ =0. when studying newtonian mechanics, it is best to remember the 1st law in two ways: if the net force (i.e., sum of all forces) acting on an object is zero, the object will keep moving with constant velocity (which may be zero). if an object is moving with constant velocity, that is, with zero acceleration, then the net force acting on that object must be zero. complete the following sentences to see if you can apply these ideas. part a if a car is moving to the left with constant velocity, one can conclude that