Let's get Mathematical Part 6

Given: cannon pinion 12 leaves, minute wheel 40 teeth, minute wheel pinion 10 leaves. How many teeth does the hour wheel have? Ratio 12:1 Motion work (teeth:pinions):

N4 = Z1•Z3
N1 Z2•Z4

12 = 12x10
1 40x?

12 = 12x1
1 4x?

So 12x12 (ratio of teeth x number of pinion leaves) = 144
Divided by the known number of teeth (4) = 36

So the answer is 36. The hour wheel has 36 teeth.

Let's get Mathematical Part 5

Calculate the maximum number of turns of a mainspring when the barrel has an interior diameter of 7.60mm, the barrel arbour, core has a diameter of 2.20mm and the mainspring thickness (strength) is 0.09mm.

Radius of interior diameter 3.8 (R)
Radius of arbour 1.1 (r)
Thickness of spring 0.09 (e)

(Square root of 2(R²+r²)) – R – r
e

(Square root of 2 x (14.44 x 1.21)) – 3.8 – 1.1
0.09

(Square root of 2 x 15.65) – 3.8 – 1.1
0.09

(Square root of 31.3) – 3.8 – 1.1
0.09

5.5946 – 3.8 – 1.1
0.09

0.6946
0.09

=

7.718

Let's get Mathematical Part 4

Calculate the power reserve of the watch described yesterday.

Ratio of Barrel: Centre wheel

N1 x Z1
N2 Z2

Revolutions of Barrel x No. of Barrel teeth
Revolutions of Centre Wheel No. of leaves Centre Pinion

7 x 120
1 12
7 x 10
1 1

=

70
1

=

70

Let's get Mathematical Part 3

Calculate the vibrations for a watch with the following: barrel 120 teeth, centre wheel 80 teeth, centre pinion 12 leaves, third wheel 75 teeth, third pinion 10 leaves, fourth wheel 70 teeth, fourth pinion 10 leaves, escape wheel 15 teeth, escape wheel pinion 7 leaves, number of turns of barrel 7.

Vh = Z1•Z3•Z5•(2Ze) = N1
Z2•Z4•Z6

Vh = 80•75•70•(2 x 15) = 18,000
10•10•7

Clue: If there are 15 teeth on the escape wheel, the answer will either be 18,000 or 21, 600. If there are 20 teeth on the escape wheel, the answer will be either 28,000 or 36,000.

Let's get Mathematical Part 2

Given: winding pinion 16 teeth, crown wheel 36 teeth, ratchet wheel 54 teeth, barrel 66 teeth. How many turns of the crown are required to wind the mainspring 8 turns?

The ratio of the number of revolutions is inversely proportional to the ratio of the number of teeth.

Crown wheel teeth (36) divided by winding pinion leaves (16) = 2.25
Ratchet wheel teeth (54) divided by crown wheel teeth (36) = 1.5

Therefore…
(2.25 x 1.5 = 3.375) x 8 = 27

So 27 turns of the crown will wind the mainspring 8 turns.

Let’s get Mathematical Part 1

If six turns of a mainspring provides a power reserve of 48 hours and the centre wheel pinion has 10 leaves and a pitch diameter of 3mm, what is the pitch diameter of the barrel and how many teeth will it have?

6 turns provides 48 hours so one turn of the barrel equals 8 hours, which is equivalent to eight turns of the centre wheel. Therefore the barrel has 80 teeth, which means the pitch diameter (working with the same ratio) has a 24mm pitch angle (ratio 8/1)

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