2) A wheel turns with constant angular acceleration 0.520rad/s^2 a) How much time does it take to reach an angular velocity of 8.00rad/s starting from rest? b)Through how many revolutions does the wheel turn in this time interval? 15.38 s, 9.8 rev

**a) Time to reach angular velocity:**<br /><br />* **Given:**<br /> * Initial angular velocity (ω₀) = 0 rad/s (starting from rest)<br /> * Final angular velocity (ω) = 8.00 rad/s<br /> * Angular acceleration (α) = 0.520 rad/s²<br /><br />* **Formula:** ω = ω₀ + αt<br /><br />* **Solution:**<br /> * Rearrange the formula to solve for time (t): t = (ω - ω₀) / α<br /> * Substitute the given values: t = (8.00 rad/s - 0 rad/s) / 0.520 rad/s²<br /> * Calculate: t ≈ 15.38 s<br /><br />**b) Number of revolutions:**<br /><br />* **Given:**<br /> * Time (t) = 15.38 s<br /> * Initial angular velocity (ω₀) = 0 rad/s<br /> * Angular acceleration (α) = 0.520 rad/s²<br /><br />* **Formula:** θ = ω₀t + (1/2)αt² (where θ is the angular displacement in radians)<br /><br />* **Solution:**<br /> * Substitute the given values: θ = (0 rad/s)(15.38 s) + (1/2)(0.520 rad/s²)(15.38 s)²<br /> * Calculate: θ ≈ 61.51 radians<br /><br />* **Convert radians to revolutions:**<br /> * 1 revolution = 2π radians<br /> * Number of revolutions = θ / 2π<br /> * Number of revolutions ≈ 61.51 radians / (2π radians/revolution)<br /> * Number of revolutions ≈ 9.78 revolutions (which rounds to 9.8 revolutions)<br /><br /><br />**Final Answer:**<br /><br />a) The time taken to reach an angular velocity of 8.00 rad/s is approximately **15.38 seconds**.<br /><br />b) The wheel turns through approximately **9.8 revolutions** in this time interval.<br />