A cargo aircraft has taken off from a mountain. It is climbing at constant speed and at an angle of θ=37 above horizontal. The engines of the plane provide a thrust of F thrust =400kN and the wings provides a lift of F lift=756kN. Drag/air resistance is negligible so F thrust, F lift, and weight are the only acting forces. What is the mass of the cargo aircraft ?
Component of thrust up
= 400,000 sin 37 Newtons
Lift = 756,000 N perpendicular to course made good so
lift up component
= 756,000 cos 37 Newtons
so
m g
=9.81 m = 400,000sin37+756,000cos37
To find the mass of the cargo aircraft, we need to use the equation that relates force, mass, and acceleration. In this case, the acceleration is equal to the gravitational acceleration since the plane is climbing vertically.
The only forces acting on the plane are the thrust force and the lift force, which are both directed upward. Therefore, the net force acting on the plane is the sum of these two forces:
Net force = Thrust force + Lift force
Now, let's resolve the forces into their components:
Thrust force (F thrust) has two components: one is in the horizontal direction and the other is in the vertical direction. Since the plane is climbing at an angle of θ = 37° above horizontal, the component of thrust force in the vertical direction is:
F thrust_vertical = F thrust * sin θ
Similarly, the lift force (F lift) also has two components: one in the horizontal direction and the other in the vertical direction. Since the plane is climbing vertically, the component of the lift force in the horizontal direction is zero. Therefore, the lift force only has a vertical component equal to its magnitude:
F lift_vertical = F lift
Now, to find the weight of the aircraft, we multiply its mass (m) by the acceleration due to gravity (g). The weight is given by:
Weight = m * g
Since the aircraft is climbing at a constant speed, the net force acting on it is zero. So we can write the equation:
Net force = Weight
Applying this to the vertical components of the forces:
F thrust_vertical + F lift_vertical = Weight
Substituting the given values:
F thrust * sin θ + F lift = m * g
Now we can calculate the mass of the cargo aircraft by rearranging the equation:
m = (F thrust * sin θ + F lift) / g
Let's calculate it:
- The thrust force (F thrust) is 400 kN, which is equal to 400,000 N.
- The lift force (F lift) is 756 kN, which is equal to 756,000 N.
- The acceleration due to gravity (g) is approximately 9.8 m/s².
Plugging in these values, as well as the angle θ = 37°, into the equation:
m = (400,000 N * sin 37° + 756,000 N) / 9.8 m/s²
Calculating this expression, we find:
m ≈ 81,432 kg
Therefore, the mass of the cargo aircraft is approximately 81,432 kg.