Physical Gears and Pulleys

The physics of gears and pulleys with a specificity of class B levers in terms ofmechanical advantage. A drive force is applied to the angular axle on the left and istransferred to the axles to the right. In figure 1 the distance travelled over a period of timeis 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 systemin which the rotation is propagated on another smaller cog which yields aneven greater increase in angular velocity. In figure 3 work is done by thesecond cog and the excess energy is utilized to attain an angular velocityin 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 islabelled as d2.Notice that the output distance transferred to the second cog yields an anglegreater than θ1. The additional angle is labelled θ2.Therefore, in a fixed time interval the angular velocity of the second cog isgreater than that of the first.

Figure 2.

Expanding on figure 1, the angular velocity is further increased on cog 3; cog 1 havingthe slowest, cog 2 having in between, and cog 3 having the fastest. This is assumingthat 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 externalsystem and as a result a portion of its energy is transferred to that system whichleaves only a fraction of energy to pass on to cog 3(the red arc indicates the workdone by cog 2). Therefore, cog 3 turns less, has a lesser angular velocity due to a lowerinput energy. Note that it is still possible that cog 3 might have a higher angular velocitythan cog 2 even with its lower input energy(in this figure it doesn't). Depending on theamount of work done by cog 2, the angular velocity will decrease. Likewise, if some externalsystem does work on cog 2 then the angular velocity of cog 3 can increase. Also consideringthat all 3 cogs are interconnected, cog 1 can also receive a boost of angular velocityfrom the other cogs, by their pulling, if an external system does work on the cogsto increase their velocity.

Here are some notes regarding the relationship between angular velocity, linear velocity(aroundthe 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 withthe 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 forceresults in higher stress, but less work interval(time in between). By the rules of torquethe 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 theconverse can also be noted for lesser radius.

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

If no external force is applied then energy is conserved, and the cycle's final output could bedirected in a loop back to the cycle's input and generate a perpetual cycle(note that perpetualcycles 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 nextnode. 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 isdecreased(shear stress). Therefore, stronger material is required and more friction is requiredto prevent the belt from slipping.

Torque increases with radius increase.

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