Question 7.5: Goal Relate angular to linear variables. Problem In a compac...
Goal Relate angular to linear variables.
Problem In a compact disc player, as the read head moves out from the center of the disc, the angular speed of the disc changes so that the linear speed at the position of the head remains at a constant value of about 1.3 m/s. (a) Find the angular speed of the compact disc when the read head is at r = 2.0 cm and again at r = 5.6 cm. (b) An old-fashioned record player rotates at a constant angular speed, so the linear speed of the record groove moving under the detector (the stylus) changes. Find the linear speed of a 45.0-rpm record at points 2.0 and 5.6 cm from the center. (c) In both the CD and phonograph record, information is recorded in a continuous spiral track. Calculate the total length of the track for a CD designed to play for 1.0 h.
Strategy This problem is just a matter of substituting numbers into the appropriate equations. Part (a) requires relating angular and linear speed with Equation 7.10, v_t = rω, solving for and substituting given values. In part (b), convert from rev/min to rad/s and substitute straight into Equation 7.10 to obtain the linear speeds. In part (c), linear speed multiplied by time gives the total distance.
v_t = rω [7.10]
(a) Find the angular speed when the read head is at r = 2.0 cm and r = 5.6 cm.
Solve v_t = rω for ω and substitute the numbers. At r = 2.0 cm:
ω = \frac{v_t}{r} = \frac{1.3 m/s}{2.0 × 10^{-2} m} = 65 rad/s
Likewise, find the angular speed at r = 5.6 cm:
ω = \frac{v_t}{r} = \frac{1.3 m/s}{5.6 × 10^{-2} m} = 23 rad/s
(b) Find the linear speed in m/s of a 45.0-rpm record at points 2.0 cm and 5.6 cm from the center.
45.0 \frac{rev}{min} = 45.0 \frac{\cancel{rev}}{\cancel{min}} (\frac{2π rad}{\cancel{rev}})(\frac{1.00 \cancel{min}}{60.0 s}) = 4.71 \frac{rad}{s}
Calculate the linear speed at r = 2.0 cm:
v_t = rω = (2.0 × 10^{-2} m)(4.71 rad/s) = 0.094 m/s
Calculate the linear speed at r = 5.6 cm:
v_t = rω = (5.6 × 10^{-2} m)(4.71 rad/s) = 0.26 m/s
(c) Calculate the total length of the track for a CD designed to play for 1.0 h.
Multiply the linear speed of the read head by the time in seconds.
Δx = v_tt = (1.3 m/s)(3 600 s) = 4 700 m
Remark Notice that for the record player in part (b), even though the angular speed is constant at all points along a radial line, the tangential speed steadily increases with increasing r. The calculation for a CD in part (c) is easy only because the linear (tangential) speed is constant. It would be considerably more difficult for a record player, where the tangential speed depends on the distance from the center.