Question

Gear Q is fixed on the pulley R. If pulley P undergoes 4.5 full rotations, how many rotations will gear S undergo? 

Hint:

In complex mechanical systems, break the problem down into simple pairs of interacting components. Trace the motion from the input (Pulley P) to the output (Gear S) one step at a time, applying the correct formula for each connection type (belt, axle, or gear mesh).

Answer 1

Step 1: Use rolling contact, not just a ratio formula.
When two circles roll against each other without slipping, whether linked by a belt or meshed directly, the arc length each one turns through at the contact point must be the same for both.

Step 2: Apply this to pulleys P and R.
\[ N_P \times 2\pi r_P = N_R \times 2\pi r_R \implies 4.5 \times 40 = N_R \times 40 \implies N_R = 4.5 \]
Gear Q sits on the same shaft as pulley R, so \( N_Q = N_R = 4.5 \).

Step 3: Apply the same rolling-contact idea to gears Q and S.
\[ N_Q \times 2\pi r_Q = N_S \times 2\pi r_S \implies 4.5 \times 20 = N_S \times 40 \]

Step 4: Solve for N_S.
\[ N_S = \frac{4.5 \times 20}{40} = \frac{90}{40} = 2.25 \]
\[ \boxed{2.25} \]

Answer 2

A third way is the unitary method: work out what happens for just 1 rotation of pulley P, then scale up.

For 1 rotation of P (radius 40), pulley R (radius 40) turns \( \frac{40}{40} = 1 \) rotation, since the radii are equal. Gear Q, fixed to the same shaft as R, also turns 1 rotation.

Gear Q (radius 20) meshes with gear S (radius 40). For every 1 rotation of Q, gear S turns \( \frac{20}{40} = 0.5 \) rotation, since the smaller gear must turn more to cover the same arc length as the bigger one.

So for every 1 rotation of P, gear S turns 0.5 rotation. Scaling up to the actual 4.5 rotations of P:

\[ N_S = 4.5 \times 0.5 = 2.25 \]

So the correct answer is 2.25 rotations.