The blades in a blender rotate at a rate of 6500 rpm . When the motor is turned off during operation, the blade slow to rest in 3.0 s . What is the angular acceleration as the blades slow down? \( (-226.9) \) 3 s

To determine the angular acceleration of the blender blades as they slow down, let's follow these steps: ### **Given:** - **Initial angular velocity (\( \omega_0 \))**: 6500 revolutions per minute (rpm) - **Final angular velocity (\( \omega \))**: 0 rpm (since the blades come to rest) - **Time (\( t \))**: 3.0 seconds ### **Objective:** Calculate the **angular acceleration (\( \alpha \))**. ### **Step 1: Convert Angular Velocity from rpm to radians per second (rad/s)** 1. **Convert rpm to revolutions per second (rev/s):** \[ \omega_0 = 6500 \, \text{rpm} \times \frac{1 \, \text{minute}}{60 \, \text{seconds}} = \frac{6500}{60} \, \text{rev/s} \approx 108.333 \, \text{rev/s} \] 2. **Convert rev/s to rad/s:** \[ \omega_0 = 108.333 \, \text{rev/s} \times 2\pi \, \text{rad/rev} \approx 108.333 \times 6.28319 \, \text{rad/s} \approx 680.678 \, \text{rad/s} \] ### **Step 2: Calculate Angular Acceleration (\( \alpha \))** Angular acceleration is defined as the change in angular velocity divided by the time taken for that change. \[ \alpha = \frac{\Delta \omega}{\Delta t} \] Where: - \( \Delta \omega = \omega - \omega_0 = 0 - 680.678 \, \text{rad/s} = -680.678 \, \text{rad/s} \) - \( \Delta t = 3.0 \, \text{seconds} \) Plugging in the values: \[ \alpha = \frac{-680.678 \, \text{rad/s}}{3.0 \, \text{s}} \approx -226.893 \, \text{rad/s}^2 \] ### **Conclusion:** The angular acceleration of the blender blades as they slow down is approximately **-226.9 rad/s²**. **Answer:** The angular acceleration is approximately –226.9 rad s⁻².