Let's get Mathematical Part 12

Given: barrel 84 teeth, centre wheel 80 teeth, third wheel 75 teeth, second wheel (fourth wheel) 70 teeth, escape wheel 15 teeth, centre wheel pinion 12 leaves, third wheel pinion 10 leaves, second wheel pinion (fourth wheel) 10 leaves, escape wheel pinion 7 leaves. Calculate the number of turns an escape wheel pinion accomplishes during one turn of the centre wheel.

Z1•Z3•Z5
Z2•Z4•Z6

80•75•70
10•10•7

Answer: 600

Let's get Mathematical Part 11

Given: hour wheel 64 teeth, minute wheel 42 teeth, cannon pinion 16 leaves. Calculate the number of leaves of the pinion.

N1 = Z1 Z3
N2 Z2 Z4

1 = 16 x ?
12 64 x 48

1 x ?
4 x 48

1 x ?
192

192/12 = 16

Let's get Mathematical Part 10

Given: hour wheel 48 teeth, minute wheel pinion 10 leaves, cannon pinion 12 leaves. Calculate the number of teeth of the minute wheel.

Motion work ratio is almost always 12:1

N1 = Z1 Z3
N2 Z2 Z4

1 = 10 x 12
12 48 x ?

1 x 10
4 x ?

10
120

120/4 = 30

Let's get Mathematical Part 9

Given: minute wheel 48 teeth, minute wheel pinion 12 leaves, hour wheel 36 teeth. How many leaves does the cannon pinion have?

Motion work ratio is almost always 12:1

N1 = Z1 Z3
N2 Z2 Z4

1 = 12 x ?
12 48 x 36

1 x ?
4 x 36

1 x ?
144

144/12 = 12

Let's get Mathematical Part 8

Given: centre wheel 80 teeth, third wheel 75 teeth, fourth wheel 70 teeth, escape wheel 15 teeth, third pinion 10 leaves, fourth pinion 10 leaves, escape pinion 7 leaves. Calculate the number of vibrations per hour, per second and the frequency (in Hertz) of the pocket watch.

Vh = Z1•Z3•Z5•(2Ze) = N1
Z2•Z4•Z6

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

Let's get Mathematical Part 7

Given: barrel arbour core diameter 3mm, barrel inner diameter 9mm, mainspring thickness 0.1. Calculate the theoretical length of the mainspring.

Pi x (R²-r²)
2e

3.14 x (4.5²-1.5²)
2 x 0.1

3.14 x (20.25-2.25)
0.2

3.14 x 18
0.2

56.52
0.2

282.6 turns

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)