Problem 22 - College Physics

Question

\(\bullet\) (a) A cylinder 0.150 \(\mathrm{m}\) in diameter rotates in a lathe at 620 \(\mathrm{rpm} .\) What is the tangential speed of the surface of the cylinder? (b) The proper tangential speed for machining cast iron is about 0.600 \(\mathrm{m} / \mathrm{s} .\) At how many revolutions per minute should a piece of stock 0.0800 \(\mathrm{m}\) in diameter be rotated in a lathe to produce this tangential speed?

Step-by-Step Solution

Answer

(a) 4.87 m/s. (b) 143 rpm.

Step 1: Calculate Tangential Speed (Part a)

To find the tangential speed of a rotating cylinder, we will use the formula for tangential speed: \[ v = rac{2\pi r n}{60} \] where \( v \) is the tangential speed, \( r \) is the radius of the cylinder, and \( n \) is the angular speed in revolutions per minute (rpm). First, we convert the diameter to radius: \[ r = \frac{0.150 \, \text{m}}{2} = 0.075 \, \text{m} \] Substitute \( r = 0.075 \, \text{m}\) and \( n = 620 \, \text{rpm} \) in the formula: \[ v = \frac{2 \pi \times 0.075 \, \text{m} \times 620}{60} \] Now calculate \( v \): \[ v \approx \frac{2 \pi \times 0.075 \times 620}{60} \approx 4.87 \, \text{m/s} \] The tangential speed is approximately \(4.87 \, \text{m/s}\).

Step 2: Determine Required RPM for Tangential Speed (Part b)

To find the necessary rpm to achieve a specific tangential speed, we rearrange the tangential speed formula for \( n \): \[ n = \frac{60v}{2\pi r} \] We know the desired tangential speed \( v = 0.600 \, \text{m/s} \) and the diameter \( 0.0800 \, \text{m}\), so the radius \( r \) is: \[ r = \frac{0.0800 \, \text{m}}{2} = 0.0400 \, \text{m} \] Plug \( v = 0.600 \, \text{m/s} \) and \( r = 0.0400 \, \text{m} \) into the formula: \[ n = \frac{60 \times 0.600}{2 \pi \times 0.0400} \] Calculate \( n \): \[ n \approx \frac{60 \times 0.600}{2 \pi \times 0.0400} \approx 143.24 \] Thus, the piece of stock should be rotated at approximately \(143 \, \text{rpm}\).

Key Concepts

Understanding Circular Motion

When we talk about circular motion, we're diving into the world where objects move in a circle at a constant speed. This motion is pivotal in many mechanical and physical applications.
One of the key components of circular motion is the idea of tangential speed. This is the speed of an object as it moves along the edge of a circle. Imagine you're on a carousel; the speed at which you move around the outside edge of the carousel is your tangential speed.
In the exercise, we found the tangential speed using the formula:

• \( v = \frac{2\pi r n}{60} \)
• \( v \) represents the tangential speed.
• \( r \) is the radius of the circular path, which is half of the diameter of the cylinder.
• \( n \) is the number of revolutions per minute (rpm).

By plugging in the values, we computed the cylinder's tangential speed as approximately 4.87 m/s. This value shows how quickly the edge of the cylinder moves through space.

Delving into Rotational Kinematics

Rotational kinematics concerns the motion of objects as they rotate, including their angular speed and angular distance. It's crucial for understanding how machines with rotating parts function.
In the context of the problem, we had to manipulate the tangential speed formula to solve for the rotational speed needed to achieve a specific surface speed on the lathe. Here, we used the equation:

• \( n = \frac{60v}{2\pi r} \)
• \( n \) indicates the desired revolutions per minute.
• \( v \) is the given tangential speed.
• \( r \) is the radius for the piece of stock.

Through simple algebra, we found that to create a tangential speed of 0.600 m/s, the stock needed to rotate at approximately 143 rpm.
Rotational kinematics allows us to predict how different speeds affect the motion of machinery, which is essential in designing effective mechanical systems.

Exploring Machining Processes

Machining processes involve precise operations on metals or other materials to shape them into desired forms. This exercise addresses a common practice in machining: ensuring the correct tangential speed for efficient cutting and finishing.
Machinists often seek to control the tangential speed because it directly impacts the quality of the finish and the speed at which material is removed. For instance:

• Higher tangential speeds can lead to smoother finishes and faster material removal but may also increase tool wear.
• Lower speeds can prolong tool life but might not deliver the best surface finish.

In our exercise, determining the proper rpm for machining cast iron demonstrates how crucial these calculations are. We needed a tangential speed of 0.600 m/s for optimal machining, which is a standard requirement for processing certain materials.
This kind of analysis helps machinists and engineers choose the right settings, balancing quality, efficiency, and tool longevity.