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Gears Problem -- tray for holding the disk in a tabletop Blu-ray player

  1. Oct 20, 2014 #1
    1. The problem statement, all variables and given/known data
    The mechanism that operates the load/unload tray for holding the disk in a tabletop Blu-ray player uses nylon spur gears, a rack, and a belt drive. The gear that meshes with the rack has a module of 2.5 mm. The gears have the numbers of teeth indicated, and the two sheaves have diameters of 7 mm and 17 mm. The rack is connected to the tray that holds the disk. For the tray to move at 0.1 m/s, how fast must the motor turn?
    LzS7F11.png

    2. Relevant equations


    3. The attempt at a solution
    ##
    \omega_{4}=\omega_{5}\\
    \omega_{2}=\omega_{3}\\
    d_{1} \omega_{1} = d_{2} \omega_{2} \\
    \omega_{2} = \frac{\omega_{1}d_{1}}{d_{2}} \\
    N_{4} \omega_{4} = N_{3} \omega_{3} \\
    N_{4} \omega_{5} = N_{3} \omega_{2} \\
    N_{4} \omega_{5} = N_{3} \frac{\omega_{1}d_{1}}{d_{2}} \\
    \omega_{1}=\frac{N_{4}\omega_{5}d_{2}}{N_{3}d_{1}} \\
    ##

    I know that the module of the gear that meshes with the rack is 2.5[mm] so the module of the rack must also be 2.5[mm].
    ##
    v=0.1[m/s]=0.1[\frac{m}{s}] \times 60[\frac{s}{min}]= 6[\frac{m}{min}] \\
    v=6 [\frac{m}{rev}][\frac{rev}{min}] = 6[\frac{m}{rev}][rpm] \\
    ##

    I need help from here.

    Also, why is the units of the module [mm] and not [mm/teeth]? The formula for it is 2r/N
     
    Last edited: Oct 20, 2014
  2. jcsd
  3. Oct 20, 2014 #2
    ##
    v=0.1[m/s]=0.1[\frac{m}{s}] \times 60[\frac{s}{min}]= 6[\frac{m}{min}]\\
    v=6 [\frac{m}{rev}][\frac{rev}{min}] = x\omega_{5} \\
    x=14[\frac{tth}{rev} \times 2.5 \frac{mm}{tth}=0.035 \frac{m}{rev} \\
    v=6 [\frac{m}{rev}][\frac{rev}{min}] = 0.035 \frac{m}{rev} \times \omega_{5} \\
    \omega_{5}=171.4285714 [rpm]= 171.4285714 [rpm] \times 2\pi \frac{rad}{rev} \times \frac{1}{60} \frac{min}{s}= 17.95195802 [rad/s]
    ##

    Did I get the angular velocity correctly for the gear meshed with the rack/tray?
     
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