Due to an unequal distribution of fuel in the wing tanks,...

Due to an unequal distribution of fuel in the wing tanks, the centers of gravity for the airplane fuselage $A$ and wings $B$ and $C$ are located as shown. If these components have weights $W_{A}=45000 \mathrm{lb}, \quad W_{B}=8000 \mathrm{lb}, \quad$ and $W_{C}=6000 \mathrm{lb},$ determine the normal reactions of the wheels $D, E,$ and $F$ on the ground.

Due to an unequal distribution of fuel in the wing tanks, the centers of gravity for the airplane fuselage $A$ and wings $B$ and $C$ are located as shown. If these components have weights $W_{A}=45000 \mathrm{lb}, \quad W_{B}=8000 \mathrm{lb}, \quad$ and
$W_{C}=6000 \mathrm{lb},$ determine the normal reactions of the wheels $D, E,$ and $F$ on the ground.

Key Concepts

Free-Body Diagram

A free-body diagram is a graphical representation of all the forces acting on an individual body or a system. It is essential for visualizing the weight distribution, locations of centers of gravity, and the points at which reaction forces act, and it serves as the primary tool for setting up the equilibrium equations.

Reaction Forces

Reaction forces are the forces exerted by supports or surfaces that prevent motion of a structure. In the context of an aircraft on the ground, the normal reactions at the wheels are the vertical support forces that counteract the weights of the various components, maintaining overall equilibrium.

Center of Gravity

The center of gravity is the point where the total weight of a body can be considered to act. In structural analysis, knowing the position of the center of gravity for each component (or the combined system) simplifies the determination of the net moments and assists in establishing equilibrium conditions.

Static Equilibrium

Static equilibrium is the condition where an object remains at rest because the vector sum of all forces and the sum of all moments (torques) acting on it are zero. This principle underlies the analysis of systems such as aircraft on the ground, where the downward gravitational forces must be balanced by the upward normal reactions at the support points.

Moment Equilibrium

Moment (or torque) equilibrium refers to the state where the sum of the moments about any point is zero. In problems involving distributed weights and support reactions, taking moments allows one to solve for unknown forces by balancing the torques produced by the different weight components about a chosen pivot point.

Transcript

00:02 In this problem we have an airplane that has an unequal distribution of fuel in the wing tanks.

00:08 And the center of gravity for the airplane fuselage wings are in b or located as shown.

00:17 They have weights 45 ,000 pounds, 8 ,000 pounds, and 6 ,000 pounds.

00:25 So the fuselage has weight 45 ,000 pounds, and each of the wings, one wing has a weight of 6 ,000.

00:32 Pounds and the other has a weight of 8 ,000 pounds and that's because of the difference in fuel.

00:38 I want to determine their reaction forces at the wheels.

00:42 So we have three wheels.

00:44 So again i'm not going to try to reproduce this schematic, but if you can look or reproduce the figure schematically here.

00:53 But it's just a matter of writing down all the forces that are acting and we have six forces acting, three applied loads from weight and three reaction forces.

01:06 So and we're given a coordinate system and i'm just going to use that to write my position vectors.

01:13 And so basically what we see is that the z where the how the vertical component of these position vectors is irrelevant because all of the all of the forces are in the vertical direction.

01:28 So when we take cross products with the forces it doesn't matter what these values are.

01:34 So the first load is a distance 7 feet from the origin, and this is the weight of the fuselage.

01:47 Now we have a load that is six, four feet from four feet in the x direction from the origin minus six in the y direction.

02:00 And that is in the book that we call that point b.

02:05 So that's the weight of the right -hand wing and the weight of the left -hand wing and the position of the center of mass of the left -hand wing is farther out than this one...

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