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: Understanding the Concept:
This problem involves the transmission of motion through a system of pulleys and gears. The key principles are the relationships between the rotations of connected pulleys (via a belt) and meshed gears. The numbers on the diagram represent the radii of the pulleys and gears.
Step 2: Key Formula or Approach:


For pulleys connected by a belt: The linear speed of the belt is constant. Therefore, the product of the number of rotations (\(N\)) and the radius (\(r\)) is the same for both pulleys: \(N_1 r_1 = N_2 r_2\).
For components on the same axle: Components fixed to the same rotating shaft (like Q and R) have the same number of rotations: \(N_Q = N_R\).
For meshing gears: The linear speed at the point of contact (the pitch circle) is the same. Therefore, the product of the number of rotations (\(N\)) and the radius (\(r\)) is the same for both gears: \(N_A r_A = N_B r_B\).

Step 3: Detailed Explanation:


Rotations of Pulley R: Pulley P (radius \(r_P = 40\)) is connected to pulley R (radius \(r_R = 40\)) by a belt. The smaller pulleys with radii 20 and 10 appear to be idler/tensioner pulleys and do not affect the overall speed ratio between P and R. Using the pulley relationship: \[ N_P \cdot r_P = N_R \cdot r_R \] \[ 4.5 \cdot 40 = N_R \cdot 40 \] Solving for \(N_R\), we find: \[ N_R = 4.5 \text{ rotations} \]
Rotations of Gear Q: Gear Q is fixed on the same axle as pulley R. Therefore, it rotates at the same rate. \[ N_Q = N_R = 4.5 \text{ rotations} \]
Rotations of Gear S: Gear Q (radius \(r_Q = 20\)) is meshed with gear S (radius \(r_S = 40\)). Using the gear relationship: \[ N_Q \cdot r_Q = N_S \cdot r_S \] \[ 4.5 \cdot 20 = N_S \cdot 40 \] \[ 90 = N_S \cdot 40 \] Solving for \(N_S\): \[ N_S = \frac{90}{40} = \frac{9}{4} = 2.25 \text{ rotations} \]

Step 4: Final Answer:
Gear S will undergo 2.25 rotations.

Answer 2

Instead of solving for the rotations of R, then Q, then S one stage at a time, we can combine every stage into a single train ratio and apply it in one go.

For any two circular parts rolling against each other, whether by a belt or by direct meshing, the number of rotations is inversely proportional to the radius, so \( N_{\text{out}} = N_{\text{in}} \times \frac{r_{\text{in}}}{r_{\text{out}}} \) at each stage.

There are two stages here: pulley P to pulley R (radii 40 and 40), and gear Q, on the same shaft as R, to gear S (radii 20 and 40). Multiplying the two stage ratios together gives the overall train ratio from P to S:

\[ \text{Train ratio} = \frac{r_P}{r_R} \times \frac{r_Q}{r_S} = \frac{40}{40} \times \frac{20}{40} = 1 \times \frac{1}{2} = \frac{1}{2} \]

Applying this to the 4.5 rotations of P:

\[ N_S = 4.5 \times \frac{1}{2} = 2.25 \]

So the correct answer is 2.25 rotations.