Two large gears that are being designed as part of a large machine and are shown

Question

Two large gears that are being designed as part of a large machine and are shown in the figure below; each is free to rotate about a fixed axis through its center. The radius and moment of inertia of the smaller gear are 0.50 m and 1.2 kg · m2, respectively, and the radius and moment of inertia of the larger gear are 1.0 m and 18 kg · m2, respectively. The lever attached to the smaller gear is 1.0 m long and has a negligible mass.

(a) If a worker will typically apply a force of F = 1.0 N to the end of the lever, as shown in the picture above, what will be the angular accelerations of the two gears?

rad/s2 (smaller gear)

rad/s2 (larger gear)

(b) Another part of the machine (not shown) will apply a force tangentially to the outer edge of the larger gear to temporarily keep the gear system from rotating. What should the magnitude and direction of this force (clockwise or counterclockwise) be?

magnitude N direction

Explanation / Answer

a) for smaller gear,

Applying

Net torque = I x alpha

on smaller there will force due to larger gear F.

Net torque = 1 x 1 - ( 0.5 x F) = 1.2alpha

1 - 0.5 F = 1.2 alpha1 .........(i)

on larger gear:

(1 x F) = 18(alpha2) .....(ii)

F = 18 alpha2

and speed of touching point of gears will be same.

alpha1 x 0.5 = alpha2 x 1

alpha2 = 0.5 alpha1

hence F = 9 alpha1

putting in (i)

1 - 0.5(9 alpha1) = 1.2alpha1

alpha1 = 0.175 rad/s^2 . .....Ans(smaller gear)

for larger gear

alpha2 = 0.5 x 0.175 = 0.088 rad/s^2 .......Ans

b) now system is in equilibrium.

so (1 x 1) - (0.5 F ) = 0

F = 2 N

on larger gear,

(1 x F) - (1 x f) = 0

f = F = 2 N

in the same direction as applied force.

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