Consider a level flight turn of a propeller-driven airplane in which the airplane makes a complete circular turn in 1.6 min. If the plane's speed is 180 m/s, what is the radius of the circle?
1 m
What is the centripetal acceleration of the plane?
2 m/s2
180 m/s * 96s
=17280 m
radius = 17280/2π
=2750.2m
angular velocity
=2π/96s rad./s
=π/48 rad/s
Centripetal acceleration
=rω² m/s²
I get about 11.8 m/s²
To find the radius of the circle, we can use the formula for the speed of an object in uniform circular motion:
v = (2πr) / T
Where:
v = speed of the plane (180 m/s)
r = radius of the circle (unknown)
T = time taken to complete one full circle (1.6 min = 96 seconds)
Rearranging the formula to solve for r:
r = (v * T) / (2π)
Plugging in the given values:
r = (180 m/s * 96 s) / (2π)
Calculating:
r = 2880π m / (2π)
r = 1440 m
Therefore, the radius of the circle is 1440 meters.
To find the centripetal acceleration of the plane, we can use the formula:
a = (v^2) / r
Where:
v = speed of the plane (180 m/s)
r = radius of the circle (1440 meters)
Plugging in the given values:
a = (180 m/s)^2 / 1440 m
Calculating:
a = 32,400 m^2/s^2 / 1440 m
a ≈ 22.5 m/s^2
Therefore, the centripetal acceleration of the plane is approximately 22.5 m/s^2.
To find the radius of the circular turn, we can use the formula for centripetal acceleration. Centripetal acceleration, denoted as \(a_c\), is given by the equation:
\[a_c = \frac{v^2}{r}\]
where \(v\) is the velocity of the airplane and \(r\) is the radius of the circle.
Given that the velocity of the airplane (\(v\)) is 180 m/s, and the centripetal acceleration (\(a_c\)) is 2 m/s², we can rearrange the formula to solve for the radius (\(r\)).
\(a_c = \frac{v^2}{r}\)
Rearranging the formula:
\(r = \frac{v^2}{a_c}\)
Plugging in the values:
\(r = \frac{180^2}{2}\)
\(r = \frac{32400}{2}\)
\(r = 16200\) m
Therefore, the radius of the circular turn is 16200 m.