Problem 88 - Intermediate Algebra

Question

Meshing Gears. For design purposes, the large gear is described by the circle \(x^{2}+y^{2}=16 .\) The smaller gear is a circle centered at \((7,0)\) and tangent to the larger circle. Find the equation of the smaller gear. (PICTURER NOT COPY)

Step-by-Step Solution

Answer

The equation of the smaller gear is \((x - 7)^2 + y^2 = 9\).

Step 1: Identify the center and radius of the larger gear

The larger gear is described by the equation \(x^2 + y^2 = 16\). This equation represents a circle centered at \((0, 0)\) with a radius of \(r = \sqrt{16} = 4\).

Step 2: Determine the distance between the centers

The center of the larger gear is \((0, 0)\), and the center of the smaller gear is given as \((7, 0)\). The distance \(d\) between these centers is calculated using the distance formula: \(d = \sqrt{(7-0)^2 + (0-0)^2} = 7\).

Step 3: Establish the tangency condition

Since the smaller gear is tangent to the larger gear, the sum of their radii must equal the distance between their centers. If \(r_s\) is the radius of the smaller gear, then we have: \[r_s + 4 = 7.\] Solving for \(r_s\), we find \(r_s = 3\).

Step 4: Write the equation of the smaller gear

With the center of the smaller gear known to be \((7, 0)\) and its radius found to be 3, the equation of the smaller gear is: \((x - 7)^2 + y^2 = 3^2\), which simplifies to \((x - 7)^2 + y^2 = 9\).

Key Concepts

Gear Systems

When dealing with gear systems, it is essential to understand how gears interact. Gears are typically modeled as circles for simplicity in mathematics. A gear system often consists of two or more gears meshing together to transmit motion or achieve a reduction in output speed. In our gear problem, we have two circular gears, one larger and one smaller. The purpose is to find out how they meet or touch each other using their mathematical equations.

Each gear's motion is largely dependent on its radius. A larger gear turns slower than a smaller gear for the same input speed. This difference in size and rotation often is reflected in gear systems design where the size of the gears can affect how they function together. Understanding the gear equation is key to designing these systems effectively, so knowing how to derive these equations from geometric and algebraic principles is crucial for solving real-world engineering problems.

Tangency Condition

The tangency condition is pivotal when it comes to understanding how circles, such as gears, interact. For two circles to be tangent, the distance between their centers must equal the sum of their radii. This is known as the tangency condition, and it ensures that the circles just touch each other at one point without overlapping.

In the context of gears, the tangency condition becomes essential because if the gears were not tangent, they would either overlap (and potentially cause mechanical failure) or be too far apart to engage with each other. Here's how you can understand it better:

• If the sum of the radii is greater than the distance between centers, the circles overlap.
• If the sum of the radii is less, the circles do not meet.
• Exactly equal means the circles are tangent, which is ideal for smoothly transferring motion in gears.

This precise condition ensures efficiency and functionality in mechanical systems, thereby highlighting its importance in mechanical design.

Distance Formula

The distance formula is a fundamental mathematical tool derived from the Pythagorean theorem. It is used to calculate the distance between two points in a plane, given their coordinates. The formula is expressed as:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]In the exercise, it was used to determine the distance between the centers of both gears. Given that the center of the larger gear is \(0, 0\) and the smaller gear \(7, 0\), applying the distance formula verifies the distance \(d = 7\).

Understanding the distance formula is not just confined to gear equations but is broadly applicable in mathematics. It enables accurate calculations of distances in geometry and aids in measuring precise lengths, which is a common necessity in design and architectural fields.