The average angular velocity ($\bar{\omega}$) of a rotating object is given by the formula:

$\bar{\omega} = \frac{\Delta \theta}{\Delta t}$

Where:

• $\Delta \theta$ is the change in angular displacement
• $\Delta t$ is the change in time

Given the displacement equation $\theta = 2t + 0.5t^2$, we can find the angular displacement at $t = 10$ seconds and $t = 0$ seconds.

1. At $t = 0$: $\theta(0) = 2(0) + 0.5(0)^2 = 0$

2. At $t = 10$: $\theta(10) = 2(10) + 0.5(10)^2 = 20 + 0.5(100) = 20 + 50 = 70 \text{ rad}$

Now, calculating the average angular velocity: $\Delta \theta = \theta(10) - \theta(0) = 70 - 0 = 70 \text{ rad}$ $\Delta t = 10 - 0 = 10 \text{ s}$

So, the average angular velocity is: $\bar{\omega} = \frac{70}{10} = 7.0 \text{ rad/s}$

Thus, the correct answer is:

d. 7.0 rad/s

Would you like more details on how the displacement equation works or any other questions on this topic? Here are some follow-up questions:

1. How do you calculate instantaneous angular velocity from displacement?
2. What is the relationship between angular acceleration and the displacement equation?
3. How would the average angular velocity change if the equation were different, such as $\theta = 3t + t^2$?
4. Can angular velocity be negative? If so, what would that mean physically?
5. How do we find angular acceleration from the displacement equation?

Tip: The average angular velocity is simply the total angular displacement divided by the total time. To find instantaneous angular velocity, you would take the derivative of the displacement equation.