Problem 14 - University Physics with Modern Physics

Question

A circular saw blade 0.200 m in diameter starts from rest. In 6.00 s it accelerates with constant angular acceleration to an angular velocity of 140 rad/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 the blade turns 420 radians.

Step 1: Determine Initial Conditions

The saw blade starts from rest, so the initial angular velocity \( \omega_0 \) is 0 rad/s.

Step 2: Identify Given Values

We have a final angular velocity \( \omega \) of 140 rad/s, an initial angular velocity \( \omega_0 \) of 0 rad/s, and a time interval \( t \) of 6.00 s.

Step 3: Apply Angular Acceleration Formula

The formula for constant angular acceleration \( \alpha \) is \( \alpha = \frac{\omega - \omega_0}{t} \).\[ \alpha = \frac{140 - 0}{6.00} \]\[ \alpha = \frac{140}{6.00} = 23.33 \text{ rad/s}^2 \]

Step 4: Calculate the Angle Turned

Use the angular displacement formula \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \). Since \( \omega_0 = 0 \), the equation simplifies to \( \theta = \frac{1}{2} \alpha t^2 \).\[ \theta = \frac{1}{2} \times 23.33 \times (6.00)^2 \]\[ \theta = \frac{1}{2} \times 23.33 \times 36 \]\[ \theta = 420 \text{ radians} \]

Key Concepts

Angular Acceleration

Angular acceleration describes how quickly an object's angular velocity changes over time. It is similar to linear acceleration, but it applies to rotational motion. When a circular saw blade spins, it starts to rotate faster and faster, which results in an increase in angular velocity. Angular acceleration tells us how fast this increase happens.

To calculate angular acceleration, the formula used is:

• \( \alpha = \frac{\omega - \omega_0}{t} \)

where:

• \( \alpha \) is the angular acceleration in radians per second squared (rad/s²)
• \( \omega \) is the final angular velocity
• \( \omega_0 \) is the initial angular velocity
• \( t \) is the time taken to change the velocity

This measure shows us how much the angular velocity increases per second. In this exercise, the saw blade takes 6 seconds to reach an angular velocity of 140 rad/s from rest. So, its angular acceleration is calculated as \( 23.33 \text{ rad/s}^2 \).

Angular Velocity

Angular velocity is the speed of rotation of an object and is measured in radians per second (rad/s). While linear velocity looks at how fast something moves along a path, angular velocity measures how fast something spins or rotates.

When you consider the saw blade, imagine it as if you're watching a fan blade. You can tell how fast it spins, and that's what angular velocity describes.

• It starts from rest, meaning its initial angular velocity, \( \omega_0 \), is 0 rad/s.
• It reaches a final velocity of 140 rad/s.

Understanding how quickly the saw blade achieves this speed is crucial. We calculate angular acceleration with this information and confirm how intensely the blade has been set in motion. The 140 rad/s tells us exactly how fast the blade is spinning when it stops accelerating.

Angular Displacement

Angular displacement pertains to the angle through which an object rotates during its motion. It's similar to how linear displacement describes how far an object travels, but with rotations.

In the exercise, once we found the angular acceleration, we needed to determine how far, in terms of rotation, the saw blade has moved. To do this, we use:

• \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \)

where:

• \( \theta \) is the angular displacement, measured in radians
• \( \alpha \) is the angular acceleration
• \( t \) is the time
• \( \omega_0 \) is the initial angular velocity

Since the blade starts from rest, the initial angular velocity \( \omega_0 \) is zero, simplifying the formula. Calculating gives us the total angle through which the blade has turned, which is 420 radians. This measure gives a clear picture of the blade’s overall motion during the acceleration phase.