91Ó°ÊÓ

Three vertical forces act on an airplane when it is flying at a constant altitude und with a constant velocity. These ure the weight of the airplane, an aerodynamic force on the wing of the airplane, and an aerodynamic force on the airplane's horizontal tail. (The aerodynamic forces are exerted by the surrounding air, and are reactions to the forces that the wing and tail exert on the air as the aerodynamic forces are exerted by the surrounding air, and are reactions to the forces that the wing and tail exert on the air as the airplane flies through it) For a particular light airplane with a weight of 6700 \(\mathrm{N}\) , the center of gravity is 0.30 \(\mathrm{m}\) m front of the point where the wing's vertical aerodynamic force acts and 3.66 \(\mathrm{m}\) in front of the point where the tail's vertical aerodynamic force acts. Determine the magnitude and direction (upward or down-ward) of each of the two vertical aerodvamic forces.

Step by step solution

01

Understand the Equilibrium Conditions

Since the airplane is flying at a constant altitude and velocity, the net vertical force on it must be zero. This means the sum of the aerodynamic forces and the weight of the airplane must equal zero for vertical equilibrium. Furthermore, there is no rotational acceleration, so the sum of moments about any point should also be zero.

02

Set Up the Force Equilibrium Equation

Let \( F_w \) be the aerodynamic force on the wings and \( F_t \) be the aerodynamic force on the tail. The weight of the airplane is given as 6700 N acting downward. The equation for vertical force equilibrium is:\[F_w + F_t = 6700\,\text{N}.\]

03

Set Up the Moment Equilibrium Equation

Choose the center of gravity as the pivot point for calculating moments. The moment arm for \( F_w \) is 0.30 m (tail forward relative to center of gravity) and for \( F_t \) is 3.66 m (much further back from the center of gravity). The moment equilibrium equation about the center of gravity is:\[0.30F_w = 3.66F_t.\]

04

Solve the Moment Equation for One Force

Solve the moment equation for \( F_w \):\[F_w = \frac{3.66}{0.30}F_t = 12.2F_t.\]

05

Substitute Back to Solve for Forces

Substitute \( F_w = 12.2F_t \) into the force equilibrium equation:\[12.2F_t + F_t = 6700,\]which simplifies to:\[13.2F_t = 6700.\]Solving for \( F_t \) gives:\[F_t = \frac{6700}{13.2} \approx 507.58\,\text{N}.\]

06

Compute the Second Force

Using \( F_t \) found in the previous step, substitute back to find \( F_w \):\[F_w = 12.2 \times 507.58 \approx 6191.68\,\text{N}.\]

07

Determine Directions of Forces

Since the aerodynamic forces and weight sum to zero and both aerodynamic forces must be upward to counteract the downward weight of the airplane, both \( F_w \) and \( F_t \) act upward.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Airplane Mechanics

Airplane mechanics involve the principles of physics that allow airplanes to fly. These include analyzing the forces and moments at play, such as lift, weight, thrust, and drag.
For the problem at hand, the focus is primarily on the forces acting vertically on the airplane during flight. When an airplane is flying at constant altitude and velocity, it's in a state of mechanical equilibrium where specific forces are perfectly balanced.

• Lift: Generated by the wings to counteract the airplane's weight.
• Weight: The gravitational force pulling the airplane downward.
• Thrust: The force propelling the airplane forward.
• Drag: The force resisting the airplane’s motion through air.

In the context of this problem, we consider the lift generated by two main aerodynamic forces: the forces on the wings and the horizontal tail, both of which must be analyzed in equilibrium conditions to ensure stable flight.

Force Equilibrium

Force equilibrium requires that all vertical forces acting on the airplane be balanced. This means that the total upward aerodynamic forces must equal the downward force of gravity (i.e., the airplane's weight) for straight and level flight.

For the airplane in the problem, this equilibrium condition can be expressed with the equation:\[F_w + F_t = 6700 \, \text{N}\]Here, \( F_w \) is the vertical aerodynamic force on the wings and \( F_t \) is the vertical aerodynamic force on the tail.
This simple yet vital equation ensures that the lift derived from aerodynamic forces exactly balances the airplane's weight, maintaining its altitude.

Moment Equilibrium

Moment equilibrium involves ensuring that the moments (or turning forces) around any pivot point are balanced. For the airplane, it's crucial that these moments balance to avoid pitching movements that could change the flight path.

By choosing the airplane's center of gravity as a point of reference, moments due to the wing and tail forces are set up as follows:

• The moment arm of the wings' force \( F_w \) relative to the center of gravity is \( 0.30 \, \text{m} \).
• The moment arm of the tail's force \( F_t \) is \( 3.66 \, \text{m} \).

To maintain moment equilibrium around the center of gravity, the following must be true:\[0.30 \, F_w = 3.66 \, F_t\]This equation implies that the relatively small aerodynamic force on the tail is compensated by a longer moment arm, balancing the moments around the pivot.

Understanding Vertical Forces

Vertical forces are essential in maintaining an airplane's altitude during flight. The key vertical force is the airplane's weight, acting downward due to gravity.

The lift generated by the wings and tail must act upwards to counterbalance this weight.
In this problem, we solve for these aerodynamic forces such that they perfectly balance with the gravitational force of 6700 N. This ensures that the aircraft neither ascends nor descends but maintains a consistent altitude.

• Aerodynamic force on wings: Calculated to be approximately 6191.68 N, acting upwards.
• Aerodynamic force on tail: Computed to be around 507.58 N, also acting upwards.

These precise calculations ensure that the airplane is in vertical force equilibrium, a key factor for steady flight conditions.