An airplane flies clockwise around an airport control tower at a speed of 190 m/s. If that control tower is 8 km from the airplane at all times, what is the plane’s angular velocity (magnitude and direction)?
To find the plane's angular velocity, we can use the formula:
ω = v / r
where:
ω is the angular velocity,
v is the linear velocity,
r is the radius.
Given:
v = 190 m/s (linear velocity),
r = 8 km = 8000 m (radius).
Substituting the given values into the formula:
ω = 190 m/s / 8000 m
Simplifying the expression:
ω = 0.02375 rad/s.
So, the magnitude of the plane's angular velocity is 0.02375 rad/s.
The direction of the plane's angular velocity is clockwise, as stated in the problem.
To find the airplane's angular velocity, we need to determine its rate of change of angle as it moves around the control tower. The angular velocity (ω) is defined as the change in angle (θ) over the change in time (t).
Given:
Speed of the airplane (v) = 190 m/s
Distance from the airplane to the control tower (r) = 8 km = 8000 m
We can relate the linear velocity (v) and angular velocity (ω) using the formula: v = ω * r, where r is the radius of the circular path.
First, we need to convert the speed from m/s to km/s:
190 m/s = 0.19 km/s
Now, we can rearrange the formula to find the angular velocity:
ω = v / r
Substituting the values:
ω = 0.19 km/s / 8 km
Simplifying:
ω = 0.02375 s^(-1)
The magnitude of the angular velocity is approximately 0.02375 s^(-1).
Since the airplane flies clockwise, the direction of the angular velocity would be negative (-) as per the right-hand rule convention. Therefore, the plane's angular velocity is approximately -0.02375 s^(-1) (clockwise).