Question

6. Gear A rotates with an angular velocity of 04 = 20% rad/s, where 0, is the angular displacement of gear A, measured in radians. Gears A, B, C, and D have radii of 20 mm, 50 mm, 25 mm, and 100 mm, respectively. When 04 = 2 rad , Find: i) The angular velocity of gear A ii) The angular acceleration of gear A iii) The angular velocity of gear B iv) The angular acceleration of gear B v) The angular velocity of gear C vi) The angular acceleration of gear C vii) The angular velocity of gear D viii) The angular acceleration of gear D (30 points)

Answer 1

GIVEN :

wa = 2042

r_A = 20 mm

r_B = 50 mm

r_C = 25 mm

r_D = 100 mm

ВА = 2rad/s

Solution:

when,
ВА = 2rad/s

wa = 2042

wa = 2*22

= wa 8rads

Also
dwa 40A A dea

The relation   
ade = wdw
is derived from
a = dw dt
and

w de dt
  by eliminating dt.

ade = wdw
.................................eqn(1)

a Adea = wadwa

a doa = 2 (0A)? dwA

QA = 2(CA) 2(04)2 WA de

aA = 2(ӨA)* (40А)

3 an = 8 (@A)

3 an = 8 (@A)
, when
ВА = 2rad/s

3 над = 8 (2)

ад 64rad/s4

If two rotating bodies contact each another, then the points in contact move
along different circular paths,So the velocity and tangential components
of acceleration of the points will be the same.

velocity at point of contact of gears are same , V_A= V_{B , since}  (V = r*$\omega$)

_{Therefore ,}

V_A= V_B

ra*WA = B* WB

substituting the values r_A  ,r_B& _{$\small \omega _{A}$}

20 * 8 = 50 * WB

WB 20 * 8 50

WB 3.2rads

B سنی || امها

since both the gears are on same shaft and the shaft is rotating with angular velocity., = 3.2 rad/s

3 II 3 II B

rc *Wc = ID * WD

25 * 3.2 = 100 * WD

WD = 25 * 3.2 100

WD 0.8rad/s

_{Tangential accaleration at point of contact are same ,
ata - ate
since,   
at = *}

_therefore,

_{ata - ate}

ГАЖ ДА Гв* ав

20 * 64 = 50 * AB

20 * 64 ав 50

ав 25.6rad/s4

ав ас

since both the gears are on same shaft and the shaft is rotating with angular acceleration,  = 25.6 rad/s²

ав ас

_{similarly,
ato = ato}

rc *ac = ID * OD

25 * 25.6 = 100 * ap

OD II 25 * 25.6 100

ар 6.4rad/s4

RESULT:

ANGULAR VELOCITY OF GEAR A ,   
= wa 8rads

ANGULAR ACCELERATION OF GEAR A
ад 64rad/s4

ANGULAR VELOCITY OF GEAR B

WB 3.2rads

ANGULAR ACCELERATION OF GEAR B

ав 25.6rad/s4

ANGULAR VELOCITY OF GEAR C ,   $\omega _{C}=3.2 rad/s$

ANGULAR ACCELERATION OF GEAR C   

ac = 25.6rad/s2

ANGULAR VELOCITY OF GEAR D

WD 0.8rad/s

ANGULAR ACCELERATION OF GEAR D   

ар 6.4rad/s4