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Solved: CALC On a compact disc (CD), music is coded in a
Chapter 9, Problem 101CP(choose chapter or problem)
Problem 101CP
CALC On a compact disc (CD), music is coded in a pat-tern of tiny pits arranged in a track that spirals outward toward the rim of the disc. As the disc spins inside a CD player, the track is scanned at a constant linear speed of v = 1.25m/s. Because the radius of the track varies as it spirals outward, the angular speed of the disc must change as the CD is played. (See Exercise 9.20.) Let’s see what angular acceleration is required to keep v constant. The equation of a spiral is r(θ) = r0 + βθ, where r0 is the radius of the spiral at θ = 0 and β is a constant. On a CD, r0 is the inner radius of the spiral track. If we take the rotation direction of the CD to be positive, β must be positive so that r increases as the disc turns and θ increases. (a) When the disc rotates through a small angle dθ, the distance scanned along the track is ds = rdθ. Using the above expression for r(θ), 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. (b) Since the track is scanned at a constant linear speed v, the distance s 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. (c) Use your expression for θ(t) to find the angular velocity wz and the angular acceleration αz as functions of time. Is αz constant? (d) On a CD, the inner radius of the track is 25.0 mm, the track radius increases by 1.55 µm per revolution, and the playing time is 74.0 min. Find r0, β, and the total number of revolutions made during the playing time. (e) Using your results from parts (c) and (d), make graphs of wz (in rad/s) versus t and αz (in rad/s2) versus t between t = 0 and t = 74.0 min.
9.20. Compact Disc. A compact disc (CD) stores music in a coded pattern of tiny pits 10-7 m deep. The pits are arranged in a track that spirals outward toward the rim of the disc; the inner and outer radii of this spiral are 25.0 mm and 58.0 mm, respectively. As the disc spins inside a CD player, the track is scanned at a constant linear speed of 1.25 m/s. (a) What is the angular speed of the CD when the innermost part of the track is scanned? The outermost part of the track? (b) The maximum playing time of a CD is 74.0 min. What would be the length of the track on such a maximum-duration CD if it were stretched out in a straight line? (c) What is the average angular acceleration of a maximum- duration CD during its 74.0-min playing time? Take the direction of rotation of the disc to be positive.
Questions & Answers
QUESTION:
Problem 101CP
CALC On a compact disc (CD), music is coded in a pat-tern of tiny pits arranged in a track that spirals outward toward the rim of the disc. As the disc spins inside a CD player, the track is scanned at a constant linear speed of v = 1.25m/s. Because the radius of the track varies as it spirals outward, the angular speed of the disc must change as the CD is played. (See Exercise 9.20.) Let’s see what angular acceleration is required to keep v constant. The equation of a spiral is r(θ) = r0 + βθ, where r0 is the radius of the spiral at θ = 0 and β is a constant. On a CD, r0 is the inner radius of the spiral track. If we take the rotation direction of the CD to be positive, β must be positive so that r increases as the disc turns and θ increases. (a) When the disc rotates through a small angle dθ, the distance scanned along the track is ds = rdθ. Using the above expression for r(θ), 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. (b) Since the track is scanned at a constant linear speed v, the distance s 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. (c) Use your expression for θ(t) to find the angular velocity wz and the angular acceleration αz as functions of time. Is αz constant? (d) On a CD, the inner radius of the track is 25.0 mm, the track radius increases by 1.55 µm per revolution, and the playing time is 74.0 min. Find r0, β, and the total number of revolutions made during the playing time. (e) Using your results from parts (c) and (d), make graphs of wz (in rad/s) versus t and αz (in rad/s2) versus t between t = 0 and t = 74.0 min.
9.20. Compact Disc. A compact disc (CD) stores music in a coded pattern of tiny pits 10-7 m deep. The pits are arranged in a track that spirals outward toward the rim of the disc; the inner and outer radii of this spiral are 25.0 mm and 58.0 mm, respectively. As the disc spins inside a CD player, the track is scanned at a constant linear speed of 1.25 m/s. (a) What is the angular speed of the CD when the innermost part of the track is scanned? The outermost part of the track? (b) The maximum playing time of a CD is 74.0 min. What would be the length of the track on such a maximum-duration CD if it were stretched out in a straight line? (c) What is the average angular acceleration of a maximum- duration CD during its 74.0-min playing time? Take the direction of rotation of the disc to be positive.
ANSWER:
Solution 101CP