Problem 15 - Applied Physics

Question

The diameter of the wheel of a wheel-and-axle is \(10.0 \mathrm{~cm} .\) (a) If a force of \(475 \mathrm{~N}\) is raised by applying a force of \(142 \mathrm{~N}\), find the diameter of the axle. (b) Find the MA.

Step-by-Step Solution

Answer

The diameter of the axle is approximately 2.99 cm, and the mechanical advantage is 3.34.

Step 1: Understand the Problem

We need to solve two parts of this problem. In part (a), we are looking to find the diameter of the axle based on the forces given. In part (b), we need to find the mechanical advantage (MA) of the system.

Step 2: Recall the Formula for Mechanical Advantage

For a wheel-and-axle system, the mechanical advantage (MA) is calculated using the formula: \[MA = \frac{F_{\text{output}}}{F_{\text{input}}} = \frac{d_{ ext{wheel}}}{d_{ ext{axle}}}\]where \(F_{\text{output}}\) is the load force (force being raised), \(F_{\text{input}}\) is the effort force (force applied), \(d_{ ext{wheel}}\) is the diameter of the wheel, and \(d_{ ext{axle}}\) is the diameter of the axle.

Step 3: Input Known Values

We know that \(F_{\text{output}} = 475\,\mathrm{N}\), \(F_{\text{input}} = 142\,\mathrm{N}\), and \(d_{ ext{wheel}} = 10\,\mathrm{cm}\). We need to find \(d_{ ext{axle}}\).

Step 4: Compute the Diameter of the Axle

From the formula \(MA = \frac{d_{ ext{wheel}}}{d_{ ext{axle}}}\), we rearrange to solve for the axle diameter: \[d_{ ext{axle}} = \frac{d_{ ext{wheel}} \times F_{\text{input}}}{F_{\text{output}}} = \frac{10\,\mathrm{cm} \times 142\,\mathrm{N}}{475\,\mathrm{N}}\]Calculate this value to find the diameter of the axle.

Step 5: Calculate the Axle Diameter

Substitute the known values to find: \[d_{ ext{axle}} = \frac{10 \times 142}{475} = \frac{1420}{475} \approx 2.99\,\mathrm{cm}\]Thus, the diameter of the axle is approximately \(2.99\,\mathrm{cm}.\)

Step 6: Calculate the Mechanical Advantage (MA)

To find the mechanical advantage, use the formula:\[MA = \frac{F_{\text{output}}}{F_{\text{input}}} = \frac{475\,\mathrm{N}}{142\,\mathrm{N}}\approx 3.34\]Thus, the mechanical advantage of the system is approximately 3.34.

Key Concepts

Wheel and Axle

The wheel and axle is a simple machine that helps us change the magnitude of the force applied to make a task easier. Picture a wheel as a circular disk attached to a smaller disk, called the axle. They work together to help in moving or lifting loads efficiently. This setup decreases the amount of effort needed to do work.

Imagine trying to open a heavy door. Using only a small knob directly wouldn't be effective. But when the knob is part of a larger wheel-and-axle system, the task becomes much easier. This is due to the distribution of force achieved by the circular motion, extending the lever's length.

• The wheel's edge moves a larger distance to enhance the force applied, while the axle moves a smaller distance but increases force output.
• Such mechanisms are widely used, from windlasses for drawing water from wells to steering wheels in automobiles.

Force Calculation

Understanding how to calculate forces in a wheel and axle mechanism is vital. When we apply a smaller force at the wheel, it transforms into a larger force at the axle, due to the difference in diameters. The basic formula used for these calculations is straightforward: \[ F_{\text{output}} = \frac{d_{\text{wheel}}}{d_{\text{axle}}} \times F_{\text{input}} \]where:

• \( F_{\text{output}} \) is the force being lifted or moved.
• \( F_{\text{input}} \) is the force applied.
• \( d_{\text{wheel}} \) is the diameter of the wheel.
• \( d_{\text{axle}} \) is the diameter of the axle.

This formula lets us determine the necessary force needed and better understand the mechanics at play. This understanding aids in designing more efficient machines and tools for diverse applications.

Physics Problem Solving

Tackling physics problems, like those involving wheel and axle systems, requires a clear logical sequence of steps. The goal is to translate a real-world problem into a solvable equation using physics principles. Here's how to approach a problem:

• **Understand the Scenario:** Read through the problem thoroughly. Identify what is given and what needs solving.
• **Use Relevant Formulas:** Use appropriate formulas that relate to the concepts involved, like force calculation or mechanical advantage in this case.
• **Input Known Values:** Substitute the known values into the equations.
• **Solve for the Unknown:** Rearrange the equations to find the unknown variable.
• **Check Your Work:** Ensure the solution makes sense logically and matches the context of the problem.

Applying these problem-solving skills not only helps with textbook problems but also builds a foundation for analyzing real-world mechanical situations.

Diameter Calculation

Calculating diameters in the context of a wheel and axle involves a straightforward manipulation of the mechanical advantage formula. When given certain forces and a part of the diameter, one can easily find the missing measurement. Here's how it works:
To find the diameter of an axle when we know the wheel's diameter and the forces applied, we rearrange the formula for mechanical advantage:\[ d_{\text{axle}} = \frac{d_{\text{wheel}} \times F_{\text{input}}}{F_{\text{output}}} \]For example, with a wheel diameter of 10 cm and forces of 142 N and 475 N, plugging the values gives:\[ d_{\text{axle}} = \frac{10 \times 142}{475} \approx 2.99 \text{ cm} \]The solution shows us the relation between the components, and this principle can be applied to various mechanical setups. Understanding this process helps when designing or modifying machinery that uses wheel-and-axle structures.