Problem 11 - University Physics with Modern Physics

Question

The rotating blade of a blender turns with constant angular acceleration 1.50 rad/s \(^{2} .\) (a) How much time does it take to reach an angular velocity of 36.0 rad/s, starting from rest? (b) Through how many revolutions does the blade turn in this time interval?

Step-by-Step Solution

Answer

(a) 24.0 s, (b) 68.8 revolutions

Step 1: Understand the Problem

We have a blender's blade starting from rest with a constant angular acceleration of \(1.50 \ \text{rad/s}^2\). We want to find out (a) the time it takes to reach an angular velocity of \(36.0 \ \text{rad/s}\), and (b) the number of revolutions it completes during this time.

Step 2: Use the Angular Kinematics Equation for Time

To find the time \(t\), we use the equation for angular velocity given by \(\omega = \omega_0 + \alpha t\), where \(\omega\) is the final angular velocity, \(\omega_0\) is the initial angular velocity, and \(\alpha\) is the angular acceleration. Here, \(\omega_0 = 0\ \text{rad/s}\), \(\omega = 36.0\ \text{rad/s}\), and \(\alpha = 1.50\ \text{rad/s}^2\). So, substituting the known values, we have:\[ 36.0 = 0 + 1.50t \]Solving for \(t\): \[ t = \frac{36.0}{1.50} = 24.0\ \text{s} \]

Step 3: Use the Angular Displacement Equation

To find the angular displacement \(\theta\) (in radians), use the equation \(\theta = \omega_0 t + \frac{1}{2} \alpha t^2\). Since \(\omega_0 = 0\), it becomes:\[ \theta = \frac{1}{2} \times 1.50 \times (24.0)^2 \]Calculating this gives:\[ \theta = 0.75 \times 576.0 = 432.0\ \text{radians} \]

Step 4: Convert Angular Displacement to Revolutions

To find the number of revolutions, convert the angular displacement from radians to revolutions using the conversion factor \(2\pi\ \text{radians/revolution}\):\[ \text{Number of revolutions} = \frac{432.0}{2\pi} \approx 68.8 \]

Key Concepts

Constant Angular Acceleration

In angular kinematics, constant angular acceleration refers to a scenario where the rate of change of angular velocity remains unchanged over time. Just like linear acceleration implies a steady rate of change in velocity, angular acceleration does the same for spinning or rotating objects. This is particularly useful when dealing with objects like the blade of a blender, which spins at a uniform pace. Constant angular acceleration is denoted by the symbol \( \alpha \). In problems involving rotational motion, the value of \( \alpha \) allows us to determine other important properties of the motion, such as angular velocity.

• Equation: \( \alpha = \frac{\text{change in }\omega}{\text{change in } t} \)
• Unit: radians per second squared (rad/s²)
• Enables prediction of future angular velocity and angular displacement

Angular Velocity

Angular velocity describes how fast something is spinning. It is an essential concept in the realm of rotational motion and provides a measure of how quickly an object rotates around a specific axis. The symbol usually used for angular velocity is \( \omega \), and it is expressed in radians per second. The faster an object spins, the higher its angular velocity.In the given exercise, we start from rest, meaning the initial angular velocity \( \omega_0 \) is zero, but with constant angular acceleration, we want to reach an angular velocity of \( 36.0 \ \text{rad/s} \).

• Equation: \( \omega = \omega_0 + \alpha t \)
• Initial condition: \( \omega_0 = 0 \ \text{rad/s} \)
• Solving: \( t = \frac{\omega - \omega_0}{\alpha} \)

Angular Displacement

Angular displacement is the angle through which an object moves on a circular path. It provides a measure of the total rotation experienced by the object about its axis. In rotational motion, angular displacement is akin to linear displacement in translational motion and is usually denoted by the symbol \( \theta \). It is measured in radians.To calculate angular displacement, particularly when starting from rest, we utilize the equation: \[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]Since the initial angular velocity \( \omega_0 \) is zero in the exercise scenario, it simplifies to: \[ \theta = \frac{1}{2} \alpha t^2 \]Once calculated, this value can be extremely helpful in finding how much the object has rotated over a specified time.

• Important for understanding the scope of movement in a rotating system
• Measured in radians but can be converted to revolutions

Revolutions

Revolutions offer a practical and intuitive way of understanding rotational movements by indicating how many full circles or 360-degree turns an object has completed. For technical accuracy, since angular displacement is often calculated in radians, we need to convert radians to revolutions.This conversion is achieved with the relationship: \[ \text{Revolutions} = \frac{\theta}{2 \pi} \]Where \( \theta \) is the angular displacement in radians, and \( 2 \pi \) is the number of radians in one full revolution.In the example, with an angular displacement of \( 432.0 \) radians, the number of revolutions is approximately

• \( \frac{432.0}{2 \pi} \approx 68.8 \) revolutions
• Useful for both practical applications and theoretical analyses