an aeroplane is flying in a horizontal circle with airspeed of 480 km/h. The wings of the plane are tilted at 40 degrees to the horizontal. What is the radius of the circle and how long does it take the plane to complete one full circle?
change km/hr to m/s
Look at a vector diagram. mg is down, lift is perpendicular to the tilt angle.
You need to realize the vertical part of lift is equal to mg (so mg=Lift*sin40
That give you lift. Now, the portion of Lift keeping the plane accelerating inward is lift*cos40.
So set lift*cos40=mv^2/r, and mass divides out, so you can calulate r.
period is the final easy part:
time=2PIr/velocity
thank you so much :)
To find the radius of the circle, we can use the following formula:
radius = (airspeed^2) / (gravity * tan(theta))
Where:
- airspeed = 480 km/h = 480,000 m/h
- gravity = 9.8 m/s^2 (acceleration due to gravity)
- theta = 40 degrees = 40 * (π/180) radians
First, let's convert the airspeed from meters per hour to meters per second:
480,000 m/h * (1 h/3600 s) ≈ 133.33 m/s
Now, let's calculate the radius:
radius = (133.33^2) / (9.8 * tan(40 * π/180))
radius ≈ 615.532 meters
The radius of the circle is approximately 615.532 meters.
To find the time it takes for the plane to complete one full circle, we can use the formula:
time = circumference / airspeed
The circumference of the circle can be found using the formula:
circumference = 2 * π * radius
Let's calculate the circumference:
circumference = 2 * π * 615.532
circumference ≈ 3867.875 meters
Now, let's calculate the time:
time = 3867.875 / 133.33
time ≈ 29.02 seconds
Therefore, it takes approximately 29.02 seconds for the plane to complete one full circle.
To determine the radius of the circle, we need to use the concept of centripetal force. In this case, the centripetal force is provided by the lift generated by the wings of the airplane. The lift force can be resolved into two components: a vertical component (L sinθ) and a horizontal component (L cosθ), where θ is the tilt angle of the wings (40 degrees in this case) and L is the lift force.
The horizontal component of the lift force acts as the centripetal force required to keep the airplane moving in a circle.
To calculate the radius of the circle, we can use the following equation:
Centripetal Force = Horizontal Component of Lift Force
mv²/r = L cosθ
where m is the mass of the airplane and v is the airspeed.
Rearranging the equation, we can solve for r:
r = (mv²) / (L cosθ)
Next, we need to convert the airspeed from km/h to m/s. Since 1 km/h = 1000 m/3600 s, the airspeed in m/s can be calculated as follows:
480 km/h * (1000 m/3600 s) = 133.33 m/s (approx.)
Now, we can plug in the given values to calculate the radius of the circle:
r = (m * 133.33 m/s²) / (L * cosθ)
Note that we need the mass and lift force specific to the airplane being considered to obtain an accurate answer.
As for the time it takes for the plane to complete one circle, it is related to the circumference of the circular path and the airspeed of the plane:
Time = Circumference / Airspeed
The circumference of a circle can be calculated using the formula 2πr, where r is the radius.
Given the radius, we can now calculate the circumference:
Circumference = 2πr
Once again, plug in the given radius value into the equation to find the circumference. Then, the time to complete one circle can be determined by dividing the circumference by the airspeed:
Time = Circumference / Airspeed
It's important to note that the specific values for the mass and lift force of the airplane are necessary to obtain actual numerical values for the radius and time.