= Spur gear
A <spur gear> is a cylindrical <gear> with straight teeth parallel to its rotation axis. A conventional external pair transmits motion between parallel axes. Unlike the inclined teeth of a <helical gear>, the ideal straight-tooth geometry produces no axial thrust from the tooth-normal contact force; there are still tangential and separating loads that the mounting must carry.<KHK Gear Nomenclature>[The ABC's of Gears, Basic Guide B, §1.1, printed pp.4–5, straight spur teeth and comparison with helical gears]{ref}
For compatible <spur gears>, reference diameter is $d=mN$, where $m$ is <gear module> and $N$ is tooth count. Standard reference axis separation is $(d_1+d_2)/2$. Two external <spur gears> have signed <angular velocity> relation $\omega_2/\omega_1=-N_1/N_2$: they rotate in opposite directions. Matching <gear module> alone is insufficient; tooth profiles and <gear pressure angles> must also be compatible.
For an original example, take <gear module> $3\,\mathrm{mm}$ and tooth counts 20 and 60. Reference diameters are $60\,\mathrm{mm}$ and $180\,\mathrm{mm}$, so standard axis separation is $120\,\mathrm{mm}$. Driving the smaller <spur gear> at $600$ revolutions per minute gives the larger one $200$ revolutions per minute in the opposite direction. In a steady ideal transmission, an input <torque> of $2\,\mathrm{N\,m}$ becomes $6\,\mathrm{N\,m}$ at the output. The corresponding input <gear tangential force> is $2/0.030\approx66.7\,\mathrm N$; multiplying this same force by the output reference radius $0.090\,\mathrm m$ recovers the output <torque>.
A simple bench demonstration marks one tooth on each <spur gear> and counts complete turns, testing the three-to-one <gear ratio>. Measuring transmitted <torques> and rotational speeds additionally tests <gear efficiency>. Observing the rotation ratio alone cannot establish a tooth's allowable load or the losses caused by <friction> and <backlash>.
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