Solved GEAR RATIO OF A COMPOUND GEAR TRAIN-Worksheet B | Chegg.com - Free Printable

Explanation:
Let’s solve each part step by step.

We are given two compound gear trains. In gear systems:

- Gear ratio = (teeth on driven gear) / (teeth on driver gear)
But for a *compound* train (multiple gears on the same axle), we multiply the individual gear ratios along the path.

Also, remember:
- When two gears mesh, the speed ratio is inverse to the tooth ratio:
Speed₁ / Speed₂ = Teeth₂ / Teeth₁
- For gears on the same axle (like B and C in both diagrams), they rotate together — same speed (RPM), but their teeth counts affect how the next gear turns.

Let’s label the first configuration (Question 1):

Gear A (driver): 10 teeth
Gear B (on same axle as C): 40 teeth
Gear C (on same axle as B): 10 teeth
Gear D (driven): 40 teeth

So the power flows:
A → meshes with B → B and C rotate together → C meshes with D.

Step 1: Ratio from A to B:
Since A drives B, and they mesh:
Speed of B / Speed of A = Teeth of A / Teeth of B = 10 / 40 = 1/4
So B turns at 1/4 the speed of A.

But B and C are on the same axle → C also turns at 1/4 the speed of A.

Step 2: C drives D:
C has 10 teeth, D has 40 teeth.
Speed of D / Speed of C = Teeth of C / Teeth of D = 10 / 40 = 1/4
So D turns at 1/4 the speed of C.

Therefore, overall:
Speed of D / Speed of A = (Speed of D / Speed of C) × (Speed of C / Speed of A) = (1/4) × (1/4) = 1/16

So gear ratio (output : input) = 1 : 16
Or sometimes written as 1/16 (meaning output is slower).

But note: “Gear ratio” can be defined as input : output or output : input, depending on context. In many engineering contexts, gear ratio = driver teeth / driven teeth per stage, and for compound trains, multiply all stages.

Let’s compute using tooth counts directly:

Stage 1: A (10) drives B (40) → ratio = 10/40 = 1/4
Stage 2: C (10) drives D (40) → ratio = 10/40 = 1/4
Total gear ratio (input speed : output speed) = (1/4) × (1/4) = 1/16
So for every 1 turn of A, D turns 1/16 turn → i.e., D completes 1/16 turns per turn of A.

But the question asks:
“For every turn of gear ‘A’, how many turns will gear ‘D’ complete?” → answer: 1/16 turns.

Is this a speed multiplier? A speed multiplier would mean output > input speed. Here output is slower (1/16), so no.

If driver (A) is connected to motor at 170 rpm, then output (D) rpm = 170 × (1/16) = 10.625 rpm.

Now check second configuration (Question 2):

Gear A (driver): 20 teeth
Gear B (same axle as C): 40 teeth
Gear C (same axle as B): 10 teeth
Gear D: 40 teeth

Flow: A → B (mesh), B & C same axle, C → D (mesh)

Stage 1: A (20) drives B (40) → speed ratio B/A = 20/40 = 1/2
So B and C turn at 1/2 speed of A.

Stage 2: C (10) drives D (40) → speed ratio D/C = 10/40 = 1/4
So D turns at 1/4 of C’s speed.

Overall: D/A = (1/2) × (1/4) = 1/8

So for every 1 turn of A, D completes 1/8 turn.

Is this a speed multiplier? No — output is slower → no

Motor = 170 rpm → D rpm = 170 × (1/8) = 21.25 rpm

Double-check calculations:

Q1:
- A→B: 10/40 = 1/4
- C→D: 10/40 = 1/4
- Total: 1/16 ✔️
- 170 ÷ 16 = 10.625 ✔️

Q2:
- A→B: 20/40 = 1/2
- C→D: 10/40 = 1/4
- Total: 1/8 ✔️
- 170 ÷ 8 = 21.25 ✔️

All correct.

Final Answer:
1) 1:16 ratio (or 1/16), 1/16 turns, no, 10.625 rpm
2) 1:8 ratio (or 1/8), 1/8 turns, no, 21.25 rpm