A Ferris wheel rotates at an angular velocity of 0.29 rad/s. Starting from rest, it reaches its operating speed with an average angular acceleration of 0.039 rad/s2. How long does it take the wheel to come up to operating speed?

3

To find the time it takes for the Ferris wheel to reach its operating speed, we can use the formula:

ω = ω₀ + α * t

where:
ω is the final angular velocity (0.29 rad/s),
ω₀ is the initial angular velocity (0 rad/s since it starts from rest),
α is the angular acceleration (0.039 rad/s²), and
t is the time we need to find.

Rearranging the formula to solve for t, we have:

t = (ω - ω₀) / α

Substituting the given values into the equation:

t = (0.29 rad/s - 0 rad/s) / 0.039 rad/s²

Simplifying the equation:

t = 0.29 rad/s / 0.039 rad/s²

Finally, calculating the result:

t ≈ 7.436 seconds

Therefore, it takes approximately 7.436 seconds for the Ferris wheel to come up to its operating speed.