Problem 14 - College Physics

Question

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

Step-by-Step Solution

Answer

Angular acceleration is 23.33 rad/s²; the blade turns 419.94 radians.

Step 1: Understand the Problem

Identify what is given and what needs to be found. The blade starts from rest, has a diameter of 0.200 meters, and reaches an angular velocity of 140 rad/s in 6.00 seconds under constant angular acceleration. We need to find the angular acceleration (\(\alpha\)) and the angle (\(\theta\)) through which the blade has turned.

Step 2: Convert Linear Diameter to Radius

Convert the diameter into radius since the radius is needed for angular formulas. \[ r = \frac{d}{2} = \frac{0.200}{2} = 0.100 \text{ m} \]

Step 3: Use Angular Kinematics Formula for Angular Acceleration

Use the angular kinematics formula: \( \omega_f = \omega_i + \alpha t \). Given initial angular velocity \( \omega_i = 0 \) (rad/s) and final angular velocity \( \omega_f = 140 \) (rad/s) with time \( t = 6.00 \) (s), solve for angular acceleration \( \alpha \):\[ 140 = 0 + \alpha \times 6.00 \]\[ \alpha = \frac{140}{6.00} = 23.33 \text{ rad/s}^2 \]

Step 4: Use Formula to Find the Angle of Rotation

Use the formula for angular displacement: \( \theta = \omega_i t + \frac{1}{2} \alpha t^2 \). With \( \omega_i = 0 \) (rad/s), \( \alpha = 23.33 \) (rad/s²), solve for \( \theta \):\[ \theta = 0 \times 6.00 + \frac{1}{2} \times 23.33 \times (6.00)^2 \]\[ \theta = \frac{1}{2} \times 23.33 \times 36.00 \]\[ \theta = 419.94 \text{ rad} \]

Step 5: Conclusion

The angular acceleration is \( 23.33 \text{ rad/s}^2 \) and the blade turns through \( 419.94 \text{ radians}\).

Key Concepts

Angular Acceleration

Angular acceleration is a key concept in understanding how quickly an object's rotational speed changes over time. It's like a version of linear acceleration but for rotating objects. To find angular acceleration, use the formula:

• \( \alpha = \frac{\omega_f - \omega_i}{t} \)

In this formula, \( \omega_f \) is the final angular velocity, \( \omega_i \) is the initial angular velocity, and \( t \) is the time it takes to change from the initial to the final velocity.
Angular acceleration is measured in "radians per second squared" or \( \text{rad/s}^2 \).
When solving the exercise, the blade starts from rest, which means \( \omega_i = 0 \) rad/s. The blade reaches 140 rad/s in 6 seconds, so \( \omega_f = 140 \) rad/s and \( t = 6 \) s. Plugging these into the formula, the calculated angular acceleration is \( 23.33 \text{ rad/s}^2 \).
This means the blade's speed increases at a constant rate of 23.33 radians for every second squared.

Angular Displacement

Angular displacement is the measure of the angle through which an object has rotated during a certain period. It is the rotational equivalent of the distance traveled in linear motion. The formula for calculating angular displacement \( \theta \) when starting from rest is:

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

In cases where the object starts from rest, \( \omega_i \) is 0, simplifying our formula to only consider the term with the angular acceleration. In the exercise, with \( \alpha = 23.33 \text{ rad/s}^2 \), and \( t = 6 \) s, the blade's angular displacement turns out to be 419.94 rad.
This value is derived from the calculation:

• \( \theta = \frac{1}{2} \times 23.33 \times 36 \)

The result tells us the total angle the blade has rotated while reaching its angular velocity.

Angular Velocity

Angular velocity describes how fast an object rotates or spins around an axis. It is defined as the rate of change of angular displacement and is measured in radians per second (rad/s).
Angular velocity can be thought of as the speed of rotation. There are two types: initial angular velocity (\( \omega_i \)) and final angular velocity (\( \omega_f \)).

• In our exercise, since the blade starts from rest, the initial angular velocity \( \omega_i \) is 0 rad/s.
• After 6 seconds of applying a constant angular acceleration, the blade reaches a final angular velocity \( \omega_f \), calculated as 140 rad/s.

Angular velocity is crucial in determining how fast something turns or rotates, showcasing the direct relationship between acceleration over time and the final speed reached.
Understanding this concept is vital for solving problems involving rotational motion.