Problem 35 - Algebra 2

Question

Mechanics Gear A drives Gear B. Gear A has \(a\) teeth and speed \(r_{A}\) in revolutions per minute \((r p m) .\) Gear \(B\) has \(b\) teeth and speed \(r_{B} .\) The quantities are related by the formula \(a r_{\mathrm{A}}=b r_{\mathrm{B}}\) Gear \(\mathrm{A}\) has 60 teeth and speed 540 \(\mathrm{rpm} .\) Gear \(\mathrm{B}\) has 45 teeth. Find the speed of Gear \(\mathrm{B}\) .

Step-by-Step Solution

Answer

The speed of Gear B is 720 RPM.

Step 1: Identify the given values

First, identify and jot down the values provided in the problem. Gear A has 60 teeth and its speed is 540 RPM. Gear B has 45 teeth and its speed is what we're trying to find.

Step 2: Fill in the provided formula

Substitute the known values into the formula \(ar_{A}=br_{B}\). Here, \(a = 60, r_{A} = 540\), and \(b = 45\), and we're solving for \(r_{B}\), the speed of Gear B. We get the equation \(60 \times 540 = 45 \times r_{B}\).

Step 3: Solve for \(r_{B}\)

Rearrange the equation to solve for \(r_{B}\). This gives \(r_{B} = \frac{60 \times 540}{45}\). Calculate the equation to determine the speed of Gear B.

Key Concepts

Revolutions Per Minute (RPM)

Revolutions per minute, often abbreviated as RPM, is a unit of rotational speed that measures how many complete turns a mechanical component makes in one minute. In gear systems, RPM is crucial because it reflects how fast a gear rotates and influences the operation of connected machinery.
When dealing with gears, understanding RPM can help you project either an increase or a decrease in speed based on gear sizes and numbers of teeth. If a gear with fewer teeth is driving a gear with more teeth, the latter will turn slower, and vice versa. Keeping track of RPM is essential as it ensures efficient and effective performance of machines and equipment. Always remember that calculating RPM accurately is vital in maintaining mechanical systems properly.

• RPM helps understand and control the speed of gear-driven systems.
• It allows for precise synchronization with other mechanical components.
• RPM is essential for determining mechanical advantage and performance in systems with multiple gears.

Mechanical Advantage

In the context of gears, mechanical advantage refers to the effectiveness of a gear system in amplifying force or torque. It's all about understanding the relationship between input force and output force. Gears can change speed, torque, or direction of mechanical energy, thus providing mechanical advantage.
When observing gears, the number of teeth plays a significant role. A gear with more teeth can amplify force while reducing speed, whereas a gear with fewer teeth does the opposite - increasing speed but decreasing force. This principle is used in different applications to meet specific needs. Knowing how to calculate and use mechanical advantage allows you to optimize gear systems to work more efficiently.

• Gears can alter the rotation speed and force applied in a system.
• A larger gear driving a smaller gear increases speed but reduces torque.
• Mechanical advantage allows systems to adapt to particular operational requirements.

Algebraic Equations

An algebraic equation is a mathematical statement that uses symbols and numbers, expressing the equality between two expressions. In gear systems, they are essential for calculating relationships between gears, such as speed ratios. For example, the equation \(a \times r_{A} = b \times r_{B}\) shows the relationship between two connected gears.
This equation helps in finding unknown values by substituting known values into the equation and solving for unknown quantities. By manipulating algebraic equations, you can easily determine gear speeds and other variables.
In this context, using algebraic equations to determine the speed of Gear B involves recognizing Gear A's speed and number of teeth, then applying the formula mentioned earlier. It's a practical application of algebraic skills in solving real-world problems.

• Algebraic equations help solve for unknown variables in mechanical systems.
• They provide a straightforward method for understanding gear systems and their interactions.
• Equations in gear systems express key relationships like speed and mechanical advantage effectively.