Gear Q is fixed on the pulley R. If pulley P undergoes 4.5 full rotations, how many rotations will gear S undergo?
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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).
Step 1: Find the belt distance travelled by pulley P.
The distance a point on P's rim travels equals its circumference times its number of rotations: \(2\pi(40) \times 4.5 = 360\pi\).
Step 2: Use this to find R's rotations.
Since P and R are joined by a belt, R's rim moves through the same distance. R's radius is also 40, so \(2\pi(40) \times N_R = 360\pi\), giving \(N_R = 4.5\).
Step 3: Carry this to gear Q, then find gear S's rotations.
Q sits on the same axle as R, so \(N_Q = 4.5\). At the meshing point between Q (radius 20) and S (radius 40), the arc length each gear turns through at contact must match: \(2\pi(20)(4.5) = 2\pi(40)(N_S)\).
Step 4: Solve and state the Final Answer.
\[
N_S = \frac{20 \times 4.5}{40} = \boxed{2.25 \text{ rotations}}
\]
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Approach Solution -2
A different way to organise the same relationships is to call the unknown quantity we want, gear S's rotations, \(x\), and work backward through the train to P, rather than computing forward from P to S.
Let the number of rotations gear S makes be \(x\).
At the meshing point between Q (radius 20) and S (radius 40), the product of rotations and radius must match on both sides.
\[
N_Q \times 20 = x \times 40 \implies N_Q = 2x
\]
Gear Q sits on the same shaft as pulley R, so they always turn together.
\[
N_R = N_Q = 2x
\]
Pulley P and pulley R are joined by a belt, and both have radius 40, so the product-of-rotations-and-radius rule again applies between them.
\[
N_P \times 40 = N_R \times 40 \implies N_P = N_R = 2x
\]
We are told \(P\) undergoes 4.5 rotations, so \(N_P = 4.5\).
\[
2x = 4.5 \implies x = 2.25
\]
