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A spinning ball has angular acceleration alpha = 6t² - 2t (in rad s⁻²). At t = 0, the angular velocity is 10 rad s⁻¹ and the angular position is 4 rad. Which of the following expressions correctly gives the angular position of the ball as a function of time?

1. (3/2) t⁴ - t² + 10t
2. t⁴ / 2 - t³ / 3 + 10t + 4
3. (2/3) t⁴ - t³ / 6 + 10t + 12
4. 2t⁴ - t³ / 2 + 5t + 4

Correct answer: t⁴ / 2 - t³ / 3 + 10t + 4

Solution

Step 1: omega = integral of alpha dt = integral of (6t² - 2t) dt = 2t³ - t² + C. At t = 0, omega = 10, so C = 10. Hence omega = 2t³ - t² + 10. Step 2: theta = integral of omega dt = t⁴/2 - t³/3 + 10t + C2. At t = 0, theta = 4, so C2 = 4. Therefore theta = t⁴/2 - t³/3 + 10t + 4.

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