A general aviation aircraft, with a wing surface area of 18 square metres, flies at sea level with a speed of 180 kilometres per hour. Its lift curve is shown below. Given that lift equals weight for this aircraft and that the aircraft flies at a 8 degree angle of attack, what is its mass (in kilograms)?
2191.5
Given that lift equals weight for this aircraft, we know that:
From the given graph we can see that at an angle of attack of 8 degrees, the lift coefficient of the aircraft's wings is about 0.78. Since it flies at sea level we know that the air density is 1.225 kilogram per cubic metre. Also the wing area is given to be 18 square metres, and we can convert the airspeed to 50 metres per second. Then:
Well, well, well, looks like we have a flying math problem here! So, we have an aircraft with a wing area of 18 square meters, flying at a speed of 180 kilometers per hour and an 8 degree angle of attack. And we know that lift equals weight for this little birdie.
Now, we need to calculate its mass, right? But how...hmm, let me put on my thinking clown wig for this.
To calculate the mass, we need to find the lift. And we can do that using the lift equation:
Lift = 0.5 * (air density) * (airspeed^2) * (wing area) * (lift coefficient)
But wait, we know that lift equals weight, and if we rearrange the equation, we can solve for mass:
Mass = (Lift) / (gravity)
Gravity is around 9.8 meters per second squared, in case you were wondering.
Now, here's the fun part. We need to find the lift coefficient from the lift curve. Given that we have an 8 degree angle of attack, we need to determine the corresponding lift coefficient from the curve. Once we know that, we can substitute it into our lift equation along with the other values.
So, my friend, I apologize for not having all the specific details, but if you can provide me with the lift coefficient at an 8 degree angle of attack from the curve, I'd be more than happy to calculate the mass for you! Let's keep the math party going!
To determine the mass of the aircraft, we need to calculate the lift force acting on the aircraft. The lift force can be calculated using the lift equation:
Lift = 1/2 * ρ * V^2 * Cl * A
Where:
ρ is the air density,
V is the airspeed,
Cl is the lift coefficient, and
A is the wing surface area.
Given:
Wing surface area (A) = 18 square meters
Airspeed (V) = 180 kilometers per hour = 50 meters per second (1 kilometer per hour = 1000 meters / 3600 seconds)
Angle of attack (α) = 8 degrees
Lift coefficient (Cl) is not given.
Since the lift curve is not provided, we will use a typical Cl value for a small general aviation aircraft at a similar angle of attack, which is around 0.6.
Now, we can calculate the lift force using the lift equation:
Lift = 1/2 * ρ * V^2 * Cl * A
Assuming standard sea-level conditions, the air density (ρ) is approximately 1.225 kilograms per cubic meter.
Lift = 1/2 * 1.225 kg/m^3 * (50 m/s)^2 * 0.6 * 18 m^2
Lift = 0.5 * 1.225 kg/m^3 * 2500 m^2/s^2 * 0.6 * 18 m^2
Lift = 164.025 kg.m/s^2
Since lift equals weight, we can equate the lift force to the weight of the aircraft:
Weight = Lift
Weight = 164.025 kg.m/s^2
Finally, the mass of the aircraft can be calculated by dividing the weight by the acceleration due to gravity (9.8 m/s^2):
Mass = Weight / acceleration due to gravity
Mass = 164.025 kg.m/s^2 / 9.8 m/s^2
Mass ≈ 16.76 kilograms
Therefore, the mass of the aircraft is approximately 16.76 kilograms.
To find the mass of the aircraft, we need to use the lift equation and the given information about the aircraft.
The lift equation is:
Lift = 0.5 * rho * V^2 * S * CL
Where:
- Lift is the force opposing the weight and keeping the aircraft in the air.
- rho is the air density.
- V is the velocity of the aircraft relative to the air.
- S is the wing surface area.
- CL is the coefficient of lift.
In this case, we are given:
- Wing surface area (S) = 18 square meters.
- Velocity (V) = 180 kilometers per hour.
However, we need to convert the velocity from kilometers per hour to meters per second, as the lift equation requires SI units. The conversion is as follows:
1 kilometer per hour = 1000 meters / 3600 seconds = 0.2778 meters per second
So, V = 180 km/h * 0.2778 m/s = 50 m/s (approximately).
Now, we need to find the coefficient of lift (CL) for a given angle of attack. The lift curve shows how the CL varies with the angle of attack. In this case, we are given that the aircraft flies at an 8-degree angle of attack. We can read the corresponding CL value from the lift curve to use in the calculation.
Once we have the CL value, we can rearrange the lift equation to solve for the mass (m) of the aircraft:
Lift = Weight
m * g = 0.5 * rho * V^2 * S * CL
Where:
- m is the mass of the aircraft.
- g is the acceleration due to gravity.
By rearranging the equation, we can solve for the mass:
m = (0.5 * rho * V^2 * S * CL) / g
To calculate the mass, we also need to know the air density (rho) and the acceleration due to gravity (g). Typically, air density at sea level is about 1.225 kg/m^3, and the acceleration due to gravity is approximately 9.8 m/s^2.
Substituting all the known values into the equation, we can calculate the mass of the aircraft.