Relating the Linear and Angular Variables — Question 4

Question 4

An idler changes direction twice but cancels from the speed ratio

Three externally meshing gears A, B and C have pitch radii $2$, $3$ and $5\,\mathrm{cm}$. A drives B, which drives C. Pitch-circle contact has no slip. A turns at $+10\,\mathrm{rad/s}$.

1. Find the angular velocities of B and C.

2. Explain why changing B's radius would not change C's speed ratio to A.

Question 4 - Solution

An idler changes direction twice but cancels from the speed ratio

(a) Derivation and calculation

External no-slip gear contacts impose opposite signed rim rates: $$R_A\omega_A+R_B\omega_B=0\Rightarrow \boxed{\omega_B=-\frac23(10)=-\frac{20}{3}\ \mathrm{rad/s}}.$$ $$R_B\omega_B+R_C\omega_C=0\Rightarrow \boxed{\omega_C=-\frac35\left(-\frac{20}{3}\right)=4\ \mathrm{rad/s}}.$$

(b) Result and interpretation

Combine the two constraints before substituting numbers: $$\frac{\omega_C}{\omega_A}= \left(-\frac{R_B}{R_C}\right)\left(-\frac{R_A}{R_B}\right) =\boxed{\frac{R_A}{R_C}}.$$ The idler radius cancels. It changes the idler’s own rate and the required center spacing, while leaving the A-to-C ratio unchanged.

Physical check

Two sign reversals restore the original rotation sense for gear C.