Pigeons are bred to display a number of interesting characteristics. One breed of pigeon, the "roller," is remarkable for the fact that it does a number of backward somersaults as it drops straight down toward the ground. Suppose a roller pigeon drops from rest and free falls downward for a distance of 15 m. If the pigeon somersaults at the rate of 13 rad/s , how many revolutions has it completed by the end of its fall?
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To determine the number of revolutions completed by the roller pigeon during its fall, we need to find the total angular displacement. The angular displacement can be calculated using the formula:
θ = ω * t
where:
θ is the angular displacement,
ω is the angular velocity, and
t is the time interval.
In this case, the angular velocity is given as 13 rad/s. However, we don't have the time interval. To find the time interval, we can use the kinematic equation for vertical free fall:
h = 1/2 * g * t^2
where:
h is the height (15m in this case),
g is the acceleration due to gravity (approximately 9.8 m/s^2), and
t is the time interval.
Rearranging the equation to solve for t:
t = sqrt(2h / g)
Substituting the given values:
t = sqrt(2 * 15 / 9.8) ≈ 1.82 s
Now that we have the time interval, we can calculate the total angular displacement:
θ = ω * t = 13 rad/s * 1.82 s ≈ 23.66 rad
To convert the angular displacement into revolutions, we need to divide it by 2π (since 1 revolution is equal to 2π radians):
Number of revolutions = θ / (2π) ≈ 23.66 rad / (2π) ≈ 3.77 revolutions
Therefore, the roller pigeon completes approximately 3.77 revolutions by the end of its fall.