Condition for Equilibrium in 3D | Engineering Mechanics for Mechanical Engineering PDF Download

We now extend our principles and methods developed for two dimensional equilibrium to the case of three-dimensional equilibrium. In Art. 3/1 the general conditions for the equilibrium of a body were stated in Eqs. 3/1, which require that the resultant force and resultant couple on a body in equilibrium be zero. These two vector equations of equilibrium and their scalar components may be written as

Condition for Equilibrium in 3D | Engineering Mechanics for Mechanical Engineering

The first three scalar equations state that there is no resultant force acting on a body in equilibrium in any of the three coordinate directions. The second three scalar equations express the further equilibrium requirement that there be no resultant moment acting on the body about any of the coordinate axes or about axes parallel to the coordinate axes. These six equations are both necessary and sufficient conditions for complete equilibrium. The reference axes may be chosen arbitrarily as a matter of convenience, the only restriction being that a right-handed coordinate system should be chosen when vector notation is used. The six scalar relationships of Eqs. 3/3 are independent conditions because any of them can be valid without the others. For example, for a car which accelerates on a straight and level road in the x-direction, Newton’s second law tells us that the resultant force on the car equals its mass times its acceleration. Thus ΣFx = 0, but the remaining two force–equilibrium equations are satisfied because all other acceleration components are zero. Similarly, if the flywheel of the engine of the accelerating car is rotating with increasing angular speed about the x-axis, it is not in rotational equilibrium about this axis. Thus, for the flywheel alone, ΣMx = 0 along with ΣFx = 0, but the remaining four equilibrium equations for the flywheel would be satisfied for its mass-center axes. In applying the vector form of Eqs. 3/3, we first express each of the forces in terms of the coordinate unit vectors i, j, and k. For the first equation, ΣF = 0, the vector sum will be zero only if the coefficients of i, j, and k in the expression are, respectively, zero. These three sums, when each is set equal to zero, yield precisely the three scalar equations of equilibrium, ΣFx = 0, ΣFy = 0, and ΣFz = 0. For the second equation, ΣM

0, where the moment sum may be taken about any convenient point O, we express the moment of each force as the cross product rF, where r is the position vector from O to any point on the line of action of the force F. Thus ΣM = Σ(r*F) = 0.

When the coefficients of i, j, and k in the resulting moment equation are set equal to zero, respectively, we obtain the three scalar moment equations ΣMx = 0, ΣMy = 0, and ΣMz = 0.

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FAQs on Condition for Equilibrium in 3D - Engineering Mechanics for Mechanical Engineering

1. What is the condition for equilibrium in 3D?
Ans. The condition for equilibrium in 3D is that the net force acting on an object should be zero and the net torque acting on the object should also be zero.
2. How do you determine if an object is in equilibrium in 3D?
Ans. To determine if an object is in equilibrium in 3D, you need to analyze the forces and torques acting on the object. If the sum of all the forces acting on the object is zero and the sum of all the torques acting on the object is zero, then the object is in equilibrium.
3. Can an object be in equilibrium if there is a net force acting on it?
Ans. No, an object cannot be in equilibrium if there is a net force acting on it. In equilibrium, the net force must be zero, which means that all the forces acting on the object must cancel each other out.
4. How does the center of gravity affect equilibrium in 3D?
Ans. The center of gravity plays a crucial role in determining the equilibrium of an object in 3D. If the center of gravity is directly above the base of support, the object is stable and will remain in equilibrium. However, if the center of gravity is outside the base of support, the object will be unstable and tend to topple over.
5. What are the differences between static and dynamic equilibrium in 3D?
Ans. In static equilibrium, an object is at rest and the net force and net torque acting on it are both zero. In dynamic equilibrium, an object is in motion with a constant velocity, and the net force and net torque acting on it are both zero. Static equilibrium deals with objects at rest, while dynamic equilibrium deals with objects in motion.
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