1. What is Gear Ratio?

The gear ratio is the ratio of the number of teeth on the output gear (driven gear) to the number of teeth on the input gear (driving gear). It can be expressed as:

\[ \text{Gear Ratio} = \frac{\text{Number of Teeth on Output Gear}}{\text{Number of Teeth on Input Gear}} \]

If the output gear has more teeth than the input gear, the gear ratio is greater than 1.

If the output gear has fewer teeth than the input gear, the gear ratio is less than 1.

2. Gear Ratio and Rotational Speed

The gear ratio inversely affects the rotational speed of the output gear compared to the input gear. This relationship is given by:

\[ \text{Output Speed} = \frac{\text{Input Speed}}{\text{Gear Ratio}} \]

A higher gear ratio results in a lower output speed.

A lower gear ratio results in a higher output speed.

3. Gear Ratio and Torque

The gear ratio directly affects the torque of the output gear compared to the input gear. Torque is the rotational force, and its relationship with gear ratio is given by:

\[ \text{Output Torque} = \text{Input Torque} \times \text{Gear Ratio} \]

A higher gear ratio results in a higher output torque.

A lower gear ratio results in a lower output torque.

4. Why Does Torque Increase with Gear Ratio?

When the output gear has more teeth than the input gear:

• The output gear rotates slower than the input gear.
• The force is distributed over a larger radius, increasing the torque.

5. Practical Example

Consider a gear system with:

• Input gear teeth: 20
• Output gear teeth: 40
• Input torque: 10 N·m
• Input speed: 100 RPM

Calculations:

\[ \text{Gear Ratio} = \frac{40}{20} = 2 \]

\[ \text{Output Speed} = \frac{100 \text{ RPM}}{2} = 50 \text{ RPM} \]

\[ \text{Output Torque} = 10 \times 2 = 20 \text{ N·m} \]

Results:

• The output gear rotates at 50 RPM.
• The output torque is 20 N·m.

6. Energy Conservation

In an ideal system (no energy loss due to friction), the power remains constant:

\[ \text{Input Power} = \text{Output Power} \]

\[ \text{Input Torque} \times \text{Input Speed} = \text{Output Torque} \times \text{Output Speed} \]

7. Applications of Gear Ratio and Torque

• Automotive Transmissions: Lower gear ratios provide higher torque for starting, higher gear ratios reduce torque for high-speed cruising.
• Industrial Machinery: Gears amplify torque for lifting or reduce speed for precision tasks.
• Robotics: Gears control the speed and torque of robotic arms and joints.