Gears - The PjProblemStrings

Gears - The PjProblemStrings.
TECTechnics Classroom TECTechnics Overview

Gears The PjProblemStrings

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: S₇P₃A₃₂ (Force-Push).

The math:
Pj Problem of Interest is of type force (push).

Gears The PjProblemStrings

(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.

Gear Speed Ratio

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:
S₂ = S₁(T₁/T₂).
Where
S₁ = speed of first shaft in train
S₂ = speed of last shaft in train
T₁ = product of teeth on all drivers
T₂ = 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.

Gear Idler

(d) Assuming a multi-matter-multi-dynamic space (S₇) 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 S₇P₃A₃₂
Motions at play, rotational at effort and resistance, PjProblemStrings S₇P₄A₄₂
Rack-and-Pinion (figure 133.1(b)
Forces at play of type push. PjProblemStrings S₇P₃A₃₂
Motion at pinion, rotational, PjProblemStrings S₇P₄A₄₂
Motion at rack, linear, PjProblemStrings S₇P₄A₄₁