Question

1 Moment of inertia of a solid uniform sphere around its axis of symmetry a) What is the volume element dV of a sphere? b) Assume a constant density p MIV, calculate the moment of inertia, remember that r is measured from the rotation axis for each volume element Use the volume of a sphere to get a solution that only depends on the mass M and radius R of the sphere. c) 2) Spinning DVD On a DVD, data is encoded in small patterns arranged in a track that spirals outward toward the rim of the disc. As the disc rotates in the DVD player, the track is scanned at a constant linear speed of v = 1.25 m/s. Because the radius of the track varies as it spirals outward, the angular speed of the disc must change as the DVD is played. Let's see what angular acceleration is required to keep v constant. The equation of a spiral is r(0) + βθ, where r, is the radius of the spiral at θ 0 and β is a constant. On a DVD, is the inner radius of the spiral track. If we take the rotation direction of the DVD to be positive, B must be positive so that r increases as the disc turns and 8 increases. a) when the disc rotates through a small angle d, the distance scanned along the track is ds = r dθ Using the above expression for r(0), integrate ds to find the total distance s scanned along the track as a function of the total angle θ through which the disc has rotated. Since the track is scanned at a constant linear speed v, the distance s is found in part (a) is equal to vt. Use this to find θ as a function of time. There will be two solutions for θ; choose the positive one, and explain why this is the solution to choose. Use your expression for θ(t) to find the angular velocity ω and the angular accleration α as functions of time. Is a constant? b) c) d) On a DVD, te inner radius of the track is 250mm, the track radius increases by 1.55 μm per revolution, and the playing time is 74.0 min. Find ro, B, and the total number of revolutions made during the playing time (remember:v- 1.25 m/s)

Answer 1