Question
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
Solution
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(309 Votes)
Phoebe
Veteran · Tutor for 9 years
Answer
**a) Time to reach angular velocity:*** **Given:** * Initial angular velocity (ω₀) = 0 rad/s (starting from rest) * Final angular velocity (ω) = 8.00 rad/s * Angular acceleration (α) = 0.520 rad/s²* **Formula:** ω = ω₀ + αt* **Solution:** * Rearrange the formula to solve for time (t): t = (ω - ω₀) / α * Substitute the given values: t = (8.00 rad/s - 0 rad/s) / 0.520 rad/s² * Calculate: t ≈ 15.38 s**b) Number of revolutions:*** **Given:** * Time (t) = 15.38 s * Initial angular velocity (ω₀) = 0 rad/s * Angular acceleration (α) = 0.520 rad/s²* **Formula:** θ = ω₀t + (1/2)αt² (where θ is the angular displacement in radians)* **Solution:** * Substitute the given values: θ = (0 rad/s)(15.38 s) + (1/2)(0.520 rad/s²)(15.38 s)² * Calculate: θ ≈ 61.51 radians* **Convert radians to revolutions:** * 1 revolution = 2π radians * Number of revolutions = θ / 2π * Number of revolutions ≈ 61.51 radians / (2π radians/revolution) * Number of revolutions ≈ 9.78 revolutions (which rounds to 9.8 revolutions)**Final Answer:**a) The time taken to reach an angular velocity of 8.00 rad/s is approximately **15.38 seconds**.b) The wheel turns through approximately **9.8 revolutions** in this time interval.