CALC On a compact disc (CD), music is coded in a pattern of tins 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 υ = l.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 υ constant. The equation of a spiral is r ( θ ) = r 0 + βθ , where r 0 is the radius of the spiral at θ = 0 and β is a constant. On a CD, r 0 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 υ , the distance s found in part (a) is equal to υt . 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 ω z and the angular acceleration α z as functions of time. Is α z constant? (d) On a CD, the inner radius of the track is 25.0 mm. the track radius increases by 1.55 μ m per revolution, and the playing time is 74.0 min. Find r 0 , β , and the total number of revolutions made during the playing time, (e) Using your results from parts (c) and (d), make graphs of ω z (in rad/s) versus t and α z (in rad/s 2 ) versus t between t = 0 and t = 74.0 min.
CALC On a compact disc (CD), music is coded in a pattern of tins 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 υ = l.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 υ constant. The equation of a spiral is r ( θ ) = r 0 + βθ , where r 0 is the radius of the spiral at θ = 0 and β is a constant. On a CD, r 0 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 υ , the distance s found in part (a) is equal to υt . 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 ω z and the angular acceleration α z as functions of time. Is α z constant? (d) On a CD, the inner radius of the track is 25.0 mm. the track radius increases by 1.55 μ m per revolution, and the playing time is 74.0 min. Find r 0 , β , and the total number of revolutions made during the playing time, (e) Using your results from parts (c) and (d), make graphs of ω z (in rad/s) versus t and α z (in rad/s 2 ) versus t between t = 0 and t = 74.0 min.
CALC On a compact disc (CD), music is coded in a pattern of tins 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 υ = l.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 υ 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 υ, the distance s found in part (a) is equal to υt. 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 velocityωz 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 ωz (in rad/s) versus t and αz (in rad/s2) versus t between t = 0 and t = 74.0 min.
Definition Definition Rate of change of angular velocity. Angular acceleration indicates how fast the angular velocity changes over time. It is a vector quantity and has both magnitude and direction. Magnitude is represented by the length of the vector and direction is represented by the right-hand thumb rule. An angular acceleration vector will be always perpendicular to the plane of rotation. Angular acceleration is generally denoted by the Greek letter α and its SI unit is rad/s 2 .
The ability of the human eye to rapidly rotate was studied using contact lenses fitted with accelerometers. While a subject, whose eyeball has radius 1.25 cm, watches a moving object her eyeball rotates through 20.0 degrees in a time interval of 58.5ms.
What is the magnitude of the average angular velocity of the eye?
rad/s.
A sphere, centred on the origin, has radius 31.7 cm,
and is spinning at 3 revolutions per second with its
axis of rotation along the z-axis. A point is chosen
with position vector a = (9, -14, 27) cm on the surface
of the sphere.
a. Calculate the vector representing the angular
velocity of the sphere, w, in rad/s.
i+
(1)
k rad/s
V =
b. Calculate the velocity v of point a in units of
ms-1
i+
k ms-1
j+
Note the units given in the question.
j+
A compact disc (CD) stores music in a coded pattern of tiny pits 10−7m 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. Part AWhat is the angular speed of the CD when scanning the innermost part of the track?
Part BWhat is the angular speed of the CD when scanning the outermost part of the track? Part CThe 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?
Part DWhat 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.
Chapter 9 Solutions
University Physics with Modern Physics (14th Edition)
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