Your boss at the Cut-Rate Cuckoo Clock Company wants to save money on materials.

Question

Your boss at the Cut-Rate Cuckoo Clock Company wants to save money on materials. You are asked to reduce the balance wheel, which is cylindrical, dimensions to one-third their original values. Assume the new wheel has the same density and the same coil spring (thus the same torsion constant) as the original.

Would the frequency of the angular simple harmonic motion (SHM) of the balance wheel increase or decrease if the dimensions of the balance wheel were changed as described?

By what factor would the frequency of the angular SHM of the balance wheel change?

Explanation / Answer

(a) angular frequency will increases

(b) angular frequency w = [k/I]^1/2

Without knowing the exact moment of inertia of the cylinder (does it have spokes? is it solid?), it would take the approximate form:
I1 = (C)mr²

Assuming a material of constant density,

mass m = density /volume

m = /V = /hr2

I1 = (C)(hr²)r²
I1 = (C)(hr^4)

By reducing the dimensions by a factor of 3,
I2 = (C)[(h/3)(r/3)^4]
I2 = (C)[hr^4] / 3^5
I2 = I1 / 35

w2 = sqrt(/I2)

w2 = w1 [I1/I2]1/2

w2 = w1[1/35]1/2

w2 = 15.6 w1

w2/w1 =15.6

angular frquency will increased by a factor of 15.6

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