A rigid body subjected to three forces acting at three points is known as a three-force member. These forces must have concurrent lines of action, except for parallel forces, where the lines of action are parallel.
For example, consider a dumpster connected to a pin support at point A and a pin attached to a hydraulic cylinder at point B.

Radicals can be formed by adding a radical to a spin-paired molecule. This is typically observed with unsaturated species, where the addition of a radical across the π bond leads to the production of a new radical by dissolving the π bond. For example, the addition of a Br radical to an alkene yields a carbon-centered radical.
Similar to charge conservation in chemical reactions, spin conservation is implicit for radical reactions. Accordingly, the product formed must possess an...

Another method of radical formation is the elimination process. It is the opposite of the addition route and is driven by the instability of the radical. For example, as depicted in Figure 1, dibenzoyl peroxide yields a pair of unstable radicals upon homolysis. Given its instability, this radical spontaneously undergoes elimination via a C–C bond cleavage to form a relatively more stable phenyl radical. The mechanism involves cleavage of the bond between the α and β positions...

When organisms require the same limited resources within an environment, they may have to compete for them. Competition is a net-negative interaction. Even if two competing individuals or populations do not interact directly, the overall fitness of both competitors is lowered as a result of not having full access to the limited resource.

Two vectors can be multiplied using a scalar product or a vector product. The resultant of a scalar product is scalar, while with vector products, the resultant is a vector. These rules of the scalar or vector product between two vectors can be applied to multiple vectors to obtain meaningful combinations. The scalar triple product is the dot product of a vector with the cross product of two vectors.
The scalar triple product is the dot product of a vector with the cross product of two vectors....

A hydraulic jump is a sudden rise in fluid depth in open channels, occurring when high-velocity (supercritical) flow transitions to low-velocity (subcritical) flow. This phenomenon requires an upstream Froude number greater than 1, as flows with Fr1<1 remain subcritical, making a hydraulic jump impossible due to the need for negative head loss, which violates thermodynamic principles.The characteristics of a hydraulic jump depend on the upstream Froude number and are classified as...