You are flying to Chicago for a weekend away from the books. In your last physics class, you learned that the airflow over the wings of the plane creates a lift force, which acts perpendicular to the wings. When the plane is flying level, the upward lift force exactly balances the downward weight force. Since O'Hare is one of the busiest airports in the world, you are not surprised when the captain announces that the flight is in a holding pattern due to the heavy traffic. He informs the passengers that the plane will be flying in a circle of radius 5 mi at a speed of 440 mph and an altitude of 24,000 ft. From the safety information card, you know that the total length of the wingspan of the plane is 275 ft. From this information, estimate the banking angle of the plane relative to the horizontal.
and of course assume no friction :)
tanTheta=v^2/gr
change v to m/s, altitude to m, radius to meters
can you explain your answer more please.
To estimate the banking angle of the plane relative to the horizontal, we can use the relationship between the lift force, gravitational force, and the banking angle.
When a plane is flying in a circle, the lift force provides the necessary centripetal force to keep the plane in a curved path. In this case, the centripetal force is provided by the component of the lift force that acts towards the center of the circle.
To calculate the lift force, we can use the equation:
Lift = Weight
Since the lift force is perpendicular to the wings, we can decompose it into two components:
Lift = Lift_horizontal + Lift_vertical
In a level flight, Lift_horizontal is equal to the weight of the plane and Lift_vertical is equal to zero. However, in a banked turn, Lift_horizontal is still equal to the weight, but Lift_vertical is non-zero.
To calculate the Lift_horizontal, we can use the equation:
Lift_horizontal = Weight
Weight = mass * gravitational acceleration
To calculate the Lift_vertical, we can use the equation:
Lift_vertical = Lift * sin(banking angle)
Now, by equating the Lift_horizontal and Weight, we can find the value of the Lift_vertical.
Let's calculate the values:
Given:
Radius of the circle (r) = 5 mi
Speed of the plane (v) = 440 mph
Wingspan of the plane (w) = 275 ft
Altitude of the plane (h) = 24,000 ft
First, we need to convert the units to a consistent system. We'll use the metric system:
1 mi = 1.60934 km
1 mph = 0.44704 m/s
1 ft = 0.3048 m
So, radius = 5 mi * 1.60934 km/mi * 1000 m/km = 8.0467 km
Speed = 440 mph * 0.44704 m/s = 196.16 m/s
Wingspan = 275 ft * 0.3048 m/ft = 83.822 m
Altitude = 24,000 ft * 0.3048 m/ft = 7,315.2 m
Next, we can calculate the weight of the plane:
Weight = mass * gravitational acceleration
We'll assume the plane has a mass of 100,000 kg:
Weight = 100,000 kg * 9.8 m/s^2 = 980,000 N
Now, we can calculate the Lift_horizontal:
Lift_horizontal = Weight = 980,000 N
Finally, we can calculate the Lift_vertical:
Lift_vertical = Lift * sin(banking angle)
Lift_vertical = Lift_horizontal - Weight = 980,000 N - 980,000 N = 0 N
Since the Lift_vertical is calculated to be zero, it means that there is no vertical lift force present. This implies that the plane is actually flying level, and not in a banked turn.
Therefore, the estimated banking angle of the plane relative to the horizontal is 0 degrees.