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%?

Find the steady-state temperature at the center of the plate = /2, = /2) for sums of w from n=1 to n=19 (only odd values). find the value of n for which the change in /2, /2) is less than 1% from the value at the previous n. at this point, we consider the solution converged. does the value of the temperature at the center seem reasonable? in this part, the values of x and y are constant scalar values. you should produce a plot of the value of n on the horizontal and the estimate of the temperature at the center of the plate on the vertical. b. using the value of n (or larger) that resulted in convergence in part (a), create a contourf plot of the temperature distribution across the entire plate. use enough spatial subdivisions to create a useful plot. comment on the results. hint: you need to determine what “x” and “y” values to use to make your t matrix. i recommend using a different number of x and y values. use enough x and y values so that the plot is “smooth”.