An airplane is flying above the Earth's surface at a height of 10.5 km. The centripetal acceleration of the plane is 16.4 m/s2. What is the angular velocity of the plane moving in uniform circular motion? Take the radius of the earth to be 6400 km.
To find the angular velocity of the plane, we need to first find the radius of the circular path the plane is following above the Earth's surface.
Given:
Height of the plane above the Earth's surface, h = 10.5 km = 10,500 m
Radius of the Earth, R = 6,400 km = 6,400,000 m
To calculate the radius of the circular path, we can sum the height of the plane and the radius of the Earth:
r = R + h
r = 6,400,000 m + 10,500 m
r = 6,410,500 m
Now, we can calculate the angular velocity using the formula for centripetal acceleration:
a = ω^2 * r
Where:
a = centripetal acceleration = 16.4 m/s^2
ω = angular velocity (unknown)
r = radius of the circular path = 6,410,500 m
Rearranging the equation, we get:
ω^2 = a / r
ω = √(a / r)
Substituting the values, we have:
ω = √(16.4 m/s^2 / 6,410,500 m)
ω ≈ 0.00255 rad/s
Therefore, the angular velocity of the plane moving in uniform circular motion is approximately 0.00255 rad/s.