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%?

Search coils and credit cards. one practical way to measure magnetic field strength uses a small, closely wound coil called a search coil. the coil is initially held with its plane perpendicular to a magnetic field. the coil is then either quickly rotated a quarter-turn about a diameter or quickly pulled out of the field. (a) derive the equation relating the total charge q that flows through a search coil to the magnetic-field magnitude b. the search coil has n turns, each with area a, and the flux through the coil is decreased from its initial maximum value to zero in a time ∆t. the resistance of the coil is r, and the total charge is q = i∆t, where i is the average current induced by the change in flux. (b) in a credit card reader, the magnetic strip on the back of a credit card is rapidly “swiped” past a coil within the reader. explain, using the same ideas that underlie the operation of a search coil, how the reader can decode the information stored in the pattern of magnetization on the strip. (c) is it necessary that the credit card be “swiped” through the reader at exactly the right speed? why or why not?

Give an example for each of the following situations: 1) negative acceleration, but positive velocity 2) positive acceleration and positive velocity 3) positive acceleration, but negative velocity 4) negative acceleration and negative velocity 5) positive velocity and no acceleration

The portion of the visible spectrum that appears brightest to the human eye is approximately 560 nm in wavelength, which corresponds to yellow-green. what is the frequency of this light?