Physical Gears and Pulleys

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The physics of gears and pulleys with a specificity of class B levers in terms of mechanical advantage. A drive force is applied to the angular axle on the left and is transferred to the axles to the right. In figure 1 the distance travelled over a period of time is illustrated as an increase of angular velocity when a class B lever is applied. In figure 2 the first figure is expanded by illustrating a second cog in the system in which the rotation is propagated on another smaller cog which yields an even greater increase in angular velocity. In figure 3 work is done by the second cog and the excess energy is utilized to attain an angular velocity in the third cog which has been decreased due to the loss of energy.

Figure 1.

Let R1 be the radius of the inner circle and R2 be the radius of the outer circle. Let θ1 be the angle which is travelled over a constant time interval. The arc length is obtained by utilizing the radius multiplied by the angle and is labelled as d2. Notice that the output distance transferred to the second cog yields an angle greater than θ1. The additional angle is labelled θ2. Therefore, in a fixed time interval the angular velocity of the second cog is greater than that of the first.

Figure 2.

Expanding on figure 1, the angular velocity is further increased on cog 3; cog 1 having the slowest, cog 2 having in between, and cog 3 having the fastest. This is assuming that no work or friction is done on the cogs.

Figure 3.

Figure 3 takes into account the factor of work. Cog 2 does some work on an external system and as a result a portion of its energy is transferred to that system which leaves only a fraction of energy to pass on to cog 3(the red arc indicates the work done by cog 2). Therefore, cog 3 turns less, has a lesser angular velocity due to a lower input energy. Note that it is still possible that cog 3 might have a higher angular velocity than cog 2 even with its lower input energy(in this figure it doesn't). Depending on the amount of work done by cog 2, the angular velocity will decrease. Likewise, if some external system does work on cog 2 then the angular velocity of cog 3 can increase. Also considering that all 3 cogs are interconnected, cog 1 can also receive a boost of angular velocity from the other cogs, by their pulling, if an external system does work on the cogs to increase their velocity.

Here are some notes regarding the relationship between angular velocity, linear velocity(around the circumference), mechanical advantage in terms of a Class B lever, and stress vs strain:

Angular Velocity: Velocity of the angle; similar triangles

Linear Velocity: Velocity around the circumference

Mechanical Advantage: The ratio of the edges of the circles; directly proportional with the force being applied to the outer circle(unless explicitly noted) as in a Class B lever. Note that in a Class C lever the ratio would be reversed.

Stress vs Strain: As the force increases, distance decreases; torque. Increased force results in higher stress, but less work interval(time in between). By the rules of torque the same can be said for increasing distance and decreasing force.

Notes:

The differences in mechanical advantage are additive; energy is conserved.

The radius determines the linear velocity; greater radius is greater velocity, and the converse can also be noted for lesser radius.

In polar coordinates the farther from the center of a cylinder, the greater the displacement for equivalent time interval.

If no external force is applied then energy is conserved, and the cycle's final output could be directed in a loop back to the cycle's input and generate a perpetual cycle(note that perpetual cycles have yet to be found due to frictional constraints and energy dissipation from those frictions).

The minimum energy dissipated is due to the ambient friction such as air and mechanical material strain.

At each node energy can be dissipated by doing work. The excess energy is transferred to the next node. Therefore, later nodes experience less strain and materials can be chosen accordingly.

An increase in angular velocity is an increase in stress and strain because the interval distance is decreased(shear stress). Therefore, stronger material is required and more friction is required to prevent the belt from slipping.

Torque increases with radius increase.

Angular drive force can be evaluated using very small radius with the torque formula.