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Equilibrium in Three Dimensions
Six scalar equations are required to express the conditions for the equilibrium of a rigid body in the general three-dimensional case, those equations are listed below:
These equations can be solved for no more than six unknowns, which generally will represent reactions at supports or connections.
During solving the most of the problems, the above scalar equations will be more conveniently obtained if we first expressed in vector form the conditions for the equilibrium of the rigid body considered.
We write and express the forces ‘F’ and position vectors ‘r’ regarding scalar components and unit vectors. Next, we compute all vector products, either by direct calculation or using determinants.
We observe that as many as three unknown reaction components may be eliminated from these computations through a judicious choice of the point ‘O’. By equating to zero the coefficients of the unit vectors in each of the two relations (see the above equation), we obtain the desired scalar equations.
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FAQs on Equilibrium in 3D - Engineering Mechanics for Mechanical Engineering
| 1. What is equilibrium in 3D? |  |
Ans. Equilibrium in 3D refers to a state in which an object or system is balanced, with no net force or torque acting on it in three dimensions. This means that the object or system is not accelerating or rotating.
| 2. How is equilibrium achieved in three dimensions? | |
Ans. Equilibrium in three dimensions is achieved when the sum of all forces acting on an object or system is zero, and the sum of all torques (moments) acting on the object or system is also zero. This ensures that there is no net force or torque, resulting in a state of balance.
| 3. What are the conditions for equilibrium in three dimensions? | |
Ans. The conditions for equilibrium in three dimensions are:
- The sum of all forces acting on the object or system must be zero in each coordinate direction (x, y, and z).
- The sum of all torques (moments) acting on the object or system must be zero about each coordinate axis (x, y, and z).
| 4. How is equilibrium different in three dimensions compared to two dimensions? | |
Ans. Equilibrium in three dimensions involves balancing forces and torques in three coordinate directions (x, y, and z), whereas equilibrium in two dimensions only requires balancing forces in two coordinate directions (x and y). In three dimensions, the object or system can also experience rotational equilibrium, which is not present in two dimensions.
| 5. What are some real-life examples of equilibrium in three dimensions? | |
Ans. Some real-life examples of equilibrium in three dimensions include:
- A suspended chandelier that is not moving or rotating.
- A bridge structure that is stably supporting the weight of vehicles and pedestrians.
- A satellite in geostationary orbit, maintaining a fixed position relative to Earth.
- A skyscraper that is not tilting or collapsing under its own weight.
- A person standing upright without any external support.
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