Question

If gear a rotates with a constant angular acceleration of aa = 90 rad>s2, starting from rest, determine the time required for gear d to attain an angular velocity of 600 rpm. Also, find the number of revolutions of gear d to attain this angular velocity. Gears a, b, c, and d have radii of 15 mm, 50 mm, 25 mm, and 75 mm, respectively.

Answer 1

solution:

$Given: (let a = alpha)

aA = 90 rad/s^2

(wd) = 600 rpm = 62.831 rad/s

(w0) = 0 rpm

ra = .015 m

rb = .05 m

rc = .025 m

rd = .075

aB = aA\times(ra/rb) = 90 * (.015/.05)

aB = 27 rad/s^2

aC = aB, Therefore aC = 27 rad/s^2

aD = aC\times(rc/rb) = 27 * (.025/.075)

aD = 9 rad/s^2$$Use angular motion analysis for a constant angular Velocity

w = (w0) + aD\timest =====> t = (w - w0)/aD

t = (62.832 - 0)/9

t = 6.981s

Find Theda D using same analysis

Theda =Theda0 +(w0)t + .5(aD)t^2 = 0 + 0 + .5(aD)t^2

Theda = .5\times9\times(6.981^2)

Theda = 219.3046 rad ====> 219.3046/2pi = 34.903 rev

Theda = 34.903 rev$