1. **State the problem:** A gymnast swings through two revolutions in 1.90 seconds. We need to find the average angular velocity. 2. **Recall the formula for average angular velocity:** $$\omega_{avg} = \frac{\Delta \theta}{\Delta t}$$ where $\Delta \theta$ is the angular displacement in radians and $\Delta t$ is the time interval. 3. **Convert revolutions to radians:** One revolution equals $2\pi$ radians. So, two revolutions equal: $$\Delta \theta = 2 \times 2\pi = 4\pi \text{ radians}$$ 4. **Plug values into the formula:** $$\omega_{avg} = \frac{4\pi}{1.90}$$ 5. **Calculate the value:** $$\omega_{avg} \approx \frac{12.566}{1.90} \approx 6.61 \text{ rad/s}$$ 6. **Interpretation:** The gymnast's average angular velocity is approximately 6.61 radians per second.