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Lift coefficient
Description 
The lift coefficient is a function of the angle of attack and typically follows the trend indicated in the following figure:
In the illustrated case, the slope is approximately 1.5 per 15 degrees, which corresponds to 0.1 1/degree or 5.73 1/radian.
ID:(7148, 0)
Variation of the lift coefficient
Description
Both airplanes and birds can modify the shape of their wings. Airplanes achieve this through the use of flaps, while birds adjust the position of their primary and secondary feathers. This allows them to achieve high lift at low speeds during takeoff and landing, and a reduced lift coefficient at high speeds.
Additionally, airplanes are equipped with spoilers that assist in braking during landing.
ID:(11072, 0)
Take off
Model
The key to taking off is to modify the wing in a way that achieves sufficient lift at lower speeds, allowing for a successful takeoff on a given runway length.
Variables
Calculations
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Calculations
Variable
Given
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Equation
To be used
Equations
The lift force ($F_L$), along with the wing span ($L$), the density ($\rho$), the wing top speed factor ($c_t$), the wing bottom speed factor ($c_b$), the upper wing length ($l_t$), the bottom wing length ($l_b$), and the speed with respect to the medium ($v$), is found in
| $ F_L = \rho L ( c_b l_b - c_t l_t ) v ^2$ |
If we consider the surface that generates lift ($S_w$), given by the wing span ($L$), the upper wing length ($l_t$), and the bottom wing length ($l_b$),
| $ S_w = \displaystyle\frac{1}{2} L ( l_t + l_b )$ |
and for the coefficient of lift ($C_L$), defined as
| $ C_L = 4\displaystyle\frac{ c_t l_t - c_b l_b }{ l_t + l_b }$ |
we obtain
| $ F_L =\displaystyle\frac{1}{2} \rho S_w C_L v ^2$ |
(ID 4417)
(ID 4441)
The lift force ($F_L$) along with the density ($\rho$), the surface that generates lift ($S_w$), the coefficient of lift ($C_L$), and the speed with respect to the medium ($v$) is represented as
| $ F_L =\displaystyle\frac{1}{2} \rho S_w C_L v ^2$ |
which, along with the body mass ($m$) and the gravitational Acceleration ($g$), must be equal to:
| $ F_g = m g $ |
that is:
$\displaystyle\frac{1}{2}\rho S_wC_Lv^2=mg$
resulting in:
| $ C_L =\displaystyle\frac{2 m g }{ \rho S_w }\displaystyle\frac{1}{ v ^2}$ |
(ID 4442)
The coefficient of lift ($C_L$) is calculated with the body mass ($m$), the gravitational Acceleration ($g$), the surface that generates lift ($S_w$), the density ($\rho$), and the speed with respect to the medium ($v$) as follows:
| $ C_L =\displaystyle\frac{2 m g }{ \rho S_w }\displaystyle\frac{1}{ v ^2}$ |
Therefore, with the proportionality constant coefficient sustainability ($c$) and the angle of attack of a wing ($\alpha$),
| $ C_L = c \alpha $ |
we have
| $ \alpha =\displaystyle\frac{2 m g }{ c \rho S_w }\displaystyle\frac{1}{ v ^2}$ |
(ID 4443)
If we equate the propulsion force ($F_p$) with the resistance force ($F_W$) with the total object profile ($S_p$), the coefficient of resistance ($C_W$), the density ($\rho$), and the speed with respect to the medium ($v$) in
| $ F_W =\displaystyle\frac{1}{2} \rho S_p C_W v ^2$ |
we obtain, for a the maximum speed ($v_p$),
$F_p = \displaystyle\frac{1}{2} \rho S_w C_L v_p ^2$
which, when solved for the maximum velocity, results in
| $ v_p = \sqrt{\displaystyle\frac{2 F_p }{ \rho S_p C_W } }$ |
(ID 14507)
The speed with respect to the medium ($v$) for an airplane taking off satisfies the equation with the maximum acceleration ($a_p$), the maximum speed ($v_p$), and the takeoff time ($t$):
| $\displaystyle\frac{dv}{dt}=a_p\left[1- \left(\displaystyle\frac{v}{v_p}\right)^2\right]$ |
Upon integration, we obtain the following expression:
$\log(v_p + v) - \log(v_p - v) - \log(v_p + v_0) + \log(v_p - v_0)= \displaystyle\frac{2 a_p}{v_p} t$
If the speed with respect to the medium ($v$) is much smaller than the maximum speed ($v_p$), the logarithms can be expanded into a Taylor series, resulting in a first-order approximation:
| $ v = v_0 + a_p t $ |
(ID 14508)
Since the velocity as a function of time is given by
| $ v = v_0 + a_p t $ |
we can express velocity as the rate of change of distance with respect to time:
$\displaystyle\frac{ds}{dt} = \sqrt{2 a_p v_p t }$
This equation can be integrated, yielding the relationship between the distance traveled and time:
| $ l = v_0 t + \displaystyle\frac{1}{2} a_p t ^2$ |
(ID 14509)
(ID 14515)
Examples
(ID 15173)
The lift coefficient is a function of the angle of attack and typically follows the trend indicated in the following figure:
In the illustrated case, the slope is approximately 1.5 per 15 degrees, which corresponds to 0.1 1/degree or 5.73 1/radian.
(ID 7148)
Both airplanes and birds can modify the shape of their wings. Airplanes achieve this through the use of flaps, while birds adjust the position of their primary and secondary feathers. This allows them to achieve high lift at low speeds during takeoff and landing, and a reduced lift coefficient at high speeds.
Additionally, airplanes are equipped with spoilers that assist in braking during landing.
(ID 11072)
(ID 15186)
ID:(1464, 0)