Gears - The PjProblemStrings

**Strings (S _{i}P_{j}A_{jk}) = S_{7}P_{3}A_{32} Base Sequence = 12735 String Sequence = 12735 - 3 - 32**

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Gears - The PjProblemStrings

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*Gears* are simple machines. They are used to change the direction of motion; decrease or increase the speed of motion; and magnify or reduce applied force.

(a) Figure 133.1(a) is an illustration of an eggbeater. Gears A, B and C enable the eggbeater to do its work. Physicists say, the *gear* is a type of a lever. Classify the eggbeater as a lever.

(b) Give a range for the theoretical mechanical advantage of the eggbeater of 133.1(a).

(c) Explain how the gears of the eggbeater change the speed of motion.

(d) Figure 133.1(b) is a rack-and-pinion arrangement of gears. What is its usefulness.

(e) Imagine a *gear train* consisting of four gears: A (10 teeth), B (40 teeth) C (20 teeth) and D (10 teeth). A in mesh with B, C is rigidly fixed on the same shaft as B; C in mesh with D. What is the overall *speed ratio* of the gear train?

(f) What is an *idler* gear?

(g) Write the PjProblemStrings at play with respect to figures 133.1(a) and 133.1(b).

**The strings**:
S_{7}P_{3}A_{32} (Force-Push).
**The math**:

Pj Problem of Interest is of type *force* (push).

(a) The eggbeater is a *third class lever*. Effort is between resistance and fulcrum and effort applied is greater than resistance inorder to increase speed.

(b) Theoretical Mechanical Advantage (T.M.A) for third class levers is a fraction. So, 0 < T.M.A <1.

(c) Gear A has 32 teeth; gear B has 8 teeth and gear C has 8 teeth.

So, there are four complete revolutions of B to one revolution of A.

So, C also revolves four times since it has the same number of teeth with B.

(d) A *pinion* is the smaller of two gears meshed together. In other words, of the two gears, the *pinion* has fewer teeth. A *rack-and-pinion* gear arrangement is used to change rotary motion to linear motion.

(e) Consider figure 133.2.

Four complete revolutions of A in one complete revolution of B (as a result of respective gear teeth).

B and C fixed to same shaft, so one complete revolution of C in one complete revolution of B.

Two complete revolution of D in one complete revolution of C (as a result of respective gear teeth).

So, overall speed reduction is 1/2.

In general, gear speed formula is as follows:

S_{2} = S_{1}(T_{1}/T_{2}).

Where

S_{1} = speed of first shaft in train

S_{2} = speed of last shaft in train

T_{1} = product of teeth on all drivers

T_{2} = product of teeth on all driven gears

(f) Two external gears that are meshed rotate in opposite directions. An *idler* is a third gear placed between the two gears to make them rotate in the same direction (figure 133.3). The *idler* does not change the gear ratio in speed calculation.

(d) Assuming a multi-matter-multi-dynamic space (S_{7}) because of the dynamism of atoms of materials and the fact that there are several matter in the space.
**Eggbeater (figure 133.1(a)**

Forces at play of type *push*. PjProblemStrings S_{7}P_{3}A_{32}

Motions at play, *rotational* at effort and resistance, PjProblemStrings S_{7}P_{4}A_{42}
**Rack-and-Pinion (figure 133.1(b)**

Forces at play of type *push*. PjProblemStrings S_{7}P_{3}A_{32}

Motion at pinion, *rotational*, PjProblemStrings S_{7}P_{4}A_{42}

Motion at rack, linear, PjProblemStrings S_{7}P_{4}A_{41}

The *point* **.** is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

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