Problem 14 - Applied Physics

Question

A driver gear with 60 teeth makes \(1600 \mathrm{rpm}\). How many teeth must the driven gear have if it makes 480 rpm?

Step-by-Step Solution

Answer

The driven gear must have 200 teeth.

Step 1: Understand the Gear Ratio Formula

The gear ratio is a measure of how the rotation speed of one gear affects another. It can be expressed as: \( \text{Gear Ratio} = \frac{\text{Teeth of Driven Gear}}{\text{Teeth of Driver Gear} } \). Moreover, the relationship between the rotational speeds (in rpm) and the gear teeth is given by \( \text{Gear Ratio} = \frac{\text{Driver Gear RPM}}{\text{Driven Gear RPM}} \).

Step 2: Insert Known Values Into Gear Ratio Equation

We know the driver gear has 60 teeth and rotates at 1600 rpm, and the driven gear rotates at 480 rpm. Substitute these values into the gear ratio equation:\[ \frac{\text{Driven Gear Teeth}}{60} = \frac{1600}{480} \]

Step 3: Simplify the RPM Ratio

Calculate the ratio of the driver gear RPM to driven gear RPM:\[ \frac{1600}{480} = \frac{10}{3} \]This indicates that for every 10 rotations the driver completes, the driven completes 3.

Step 4: Solve for Driven Gear Teeth

Set up the equation using the gear ratio with teeth:\[ \frac{\text{Driven Gear Teeth}}{60} = \frac{10}{3} \]Multiply both sides by 60 to solve for the number of teeth for the driven gear:\[ \text{Driven Gear Teeth} = 60 \times \frac{10}{3} \]

Step 5: Calculate Driven Gear Teeth

Perform the multiplication:\[ \text{Driven Gear Teeth} = \frac{600}{3} = 200 \]The driven gear must have 200 teeth to achieve the desired rotational speed.

Key Concepts

Driver Gear Teeth

In mechanical systems, understanding the driver gear is fundamental. The driver gear is the first gear in the system that receives the initial force. Its number of teeth, in this case 60, plays a critical role in determining the system's gear ratio.

The number of teeth on the driver gear directly affects how many times it will rotate relative to the driven gear. More teeth can result in slower rotations for the same system input. This is beneficial in systems looking to reduce speed or increase torque.

When setting up gear systems, always begin by identifying how many teeth are present on the driver gear, as it sets the baseline for calculating the gear ratio and understanding system performance.

Driven Gear Teeth

The driven gear, receiving force from the driver gear, is essential in determining the output speed. In our example, the driven gear needs to match a specific rotational speed of 480 rpm. To achieve this, we need to calculate the correct number of teeth.

The formula for determining driven gear teeth takes the ideal gear ratio into account. In this case, it simplifies to finding how the 60 teeth on the driver gear transfer motion to the driven gear.

• The equation used: \( \frac{\text{Driven Gear Teeth}}{60} = \frac{10}{3} \)
• Solving this provides the driven gear with 200 teeth.

This calculation ensures that the speed ratio between the two gears performs as intended, crucial for maintaining the desired system functionality.

Rotational Speed

Rotational speed, often measured in revolutions per minute (rpm), tells us how quickly a gear completes one full revolution. Understanding the rotational speed is key in tuning gear systems for different applications.

In our example, the driver gear spins at 1600 rpm, while the aim for the driven gear is to rotate at 480 rpm. This substantial difference necessitates a careful calculation of the gear teeth to achieve the appropriate speed decrease.

Rotational speeds dictate whether a gear system increases speed and decreases torque or decreases speed and increases torque, impacting everything from automotive engineering to clockwork design.

RPM Ratio

The RPM ratio reveals how much faster or slower one gear rotates relative to another. It is a pivotal component in gear systems, as it outlines the relationship between the input speed (driver gear) and the output speed (driven gear).

This ratio is calculated by dividing the rpm of the driver by the rpm of the driven gear, which results in the ratio \( \frac{1600}{480} = \frac{10}{3} \).

• An RPM ratio greater than one means the driver rotates faster.
• An RPM ratio less than one indicates the driver rotates slower.

Real-world decisions, like choosing the number of teeth on the driven gear, depend heavily on understanding and utilizing this ratio effectively.

Teeth Calculation

Calculating the number of teeth required for gears involves using the gear ratio and understanding what each gear's teeth contribute to the overall system. In our specific scenario, the driver gear with 60 teeth and the RPM ratio of \( \frac{10}{3} \) guide us to the correct configuration.

The calculation is straightforward but crucial for achieving the right balance between speed and torque. This is done by multiplying the number of teeth of the driver gear by the RPM ratio:

• Equation: \( \text{Driven Gear Teeth} = 60 \times \frac{10}{3} = 200 \)

By following through with this arithmetic, you establish the perfect number of teeth on the driven gear, ensuring it performs as needed in any mechanical setup.