Problem 14 - University Physics with Modern Physics

Question

A circular saw blade 0.200 \(\mathrm{m}\) in diameter starts from rest. In 6.00 \(\mathrm{s}\) it accelerates with constant angular acceleration to an angular velocity of 140 \(\mathrm{rad} / \mathrm{s}\) . Find the angular acceleration and the angle through which the blade has turned.

Step-by-Step Solution

Answer

Angular acceleration is 23.33 rad/s² and angular displacement is 419.94 rad.

Step 1: Understanding the Problem

We are given a circular saw blade with an initial angular velocity of 0 rad/s and a final angular velocity of 140 rad/s over a time period of 6.00 seconds. We need to find the angular acceleration and the angle displaced by the blade during this time.

Step 2: Relate Angular Quantities

The relationship between angular velocity (\(\omega\)), angular acceleration (\(\alpha\)), and time (\(t\)) is given by: \(\omega_f = \omega_i + \alpha t\). Here, \(\omega_i = 0\) rad/s, \(\omega_f = 140\) rad/s and \(t = 6.00\) s.

Step 3: Calculate Angular Acceleration

Using the formula \(\omega_f = \omega_i + \alpha t\), we solve for \(\alpha\):\[ \alpha = \frac{\omega_f - \omega_i}{t} = \frac{140 \text{ rad/s} - 0 \text{ rad/s}}{6.00 \text{ s}} = \frac{140}{6.00} \approx 23.33 \text{ rad/s}^2 \].

Step 4: Find Angular Displacement

The angular displacement (\(\theta\)) can be calculated using \(\theta = \omega_i t + \frac{1}{2} \alpha t^2\). Here, \(\omega_i = 0\) and we already have \(\alpha\):\[ \theta = 0 \times 6.00 + \frac{1}{2} \times 23.33 \times (6.00)^2 = \frac{1}{2} \times 23.33 \times 36.00 \approx 419.94 \text{ rad} \].

Key Concepts

Angular Acceleration

Angular acceleration is a key concept describing how quickly the angular velocity of an object changes over time. When a circular saw blade starts from rest and speeds up to 140 rad/s in 6 seconds, it undergoes constant angular acceleration. This acceleration is denoted by the symbol \( \alpha \).

To calculate angular acceleration, you can use the formula:

• \( \omega_f = \omega_i + \alpha t \)

where \( \omega_f \) represents the final angular velocity, \( \omega_i \) the initial angular velocity, and \( t \) the time taken. In our example:

• Final velocity (\( \omega_f \)) = 140 rad/s
• Initial velocity (\( \omega_i \)) = 0 rad/s
• Time (\( t \)) = 6.00 s

Substituting these values, the angular acceleration \( \alpha \) is:

• \( \alpha = \frac{140 \text{ rad/s} - 0 \text{ rad/s}}{6.00 \text{ s}} \approx 23.33 \text{ rad/s}^2 \)

This tells us that the blade's rotation speed increased at a rate of 23.33 rad/s², which means it was gaining 23.33 rotations per second-squared.

Angular Velocity

Angular velocity indicates how fast an object rotates or spins around a particular axis. In simple terms, it's the rate at which an angle changes with respect to time and is typically measured in radians per second (rad/s). For the circular saw blade, it begins with zero angular velocity and ends at 140 rad/s after 6 seconds.

Understanding angular velocity helps in knowing the rotational speed of objects like wheels, gears, or blades. In the formula:

• \( \omega = \frac{\theta}{t} \)

\( \theta \) is the angular displacement (the angle through which something moves), and \( t \) is time. However, for changes, we often use:

• \( \omega_f = \omega_i + \alpha t \)

Since \( \omega_i \) was 0, the angular velocity after 6 seconds reflects how the angular acceleration impacted the saw blade's speed over time. Reaching 140 rad/s means it completes 140 radians of rotation every second once fully accelerated.

Angular Displacement

Angular displacement represents the total angle through which an object has rotated during its motion, measured in radians. It's crucial for understanding how much an object has turned or twisted in a given duration. For the circular saw blade example, we calculated the angular displacement using:

• \( \theta = \omega_i t + \frac{1}{2} \alpha t^2 \)

Given that the blade starts from rest (\( \omega_i = 0 \)), this simplifies to focus on the acceleration part:

• \( \theta = \frac{1}{2} \times 23.33 \times (6.00)^2 \approx 419.94 \text{ rad} \)

This result can be interpreted as the circular saw blade having rotated through just under 420 radians over the 6 seconds it was accelerating, which translates to over 60 complete rotations (considering there are about 6.28 radians in one full circle). Understanding this measurement helps in visualizing the extent of rotation achieved by the blade.