The principle of Saint-Venant postulates that the stress distribution within a structural member does not rely on the precise method of load application except in the vicinity of the load application points. Consider a scenario where loads are centrally applied on two plates. In this case, the plates move toward each other without any rotation. This movement causes the member to contract in length and expand in width and thickness. Uniform deformation across all elements and maintaining...

Archimedes' principle states that an upward buoyant force exerted on a body that is immersed partially or entirely in a fluid is equal to the weight of the fluid displaced by it. To understand how much buoyant force is needed to make an object float, let us think about what happens when a submerged object is removed from a fluid. If the object were not in the fluid, the space occupied by the object would be filled by the fluid having a weight wfl. This weight is supported by the surrounding...

Imagine a rigid body with a mass denoted as 'm', which has its center of mass at point G and is rotating around an inertial reference frame. The angular momentum at an arbitrary point P can be calculated by taking the cross product of the position vector and linear momentum vector for each individual mass element.
The velocity of a mass element comprises its translational velocity and the relative velocity instigated by the body's rotation. Substituting the velocity equation into the angular...

Newton's law of gravitation describes the gravitational force between any two point masses. However, for extended spherical objects like the Earth, the Moon, and other planets, the law holds with an assumption that masses of spherical objects are concentrated at their respective centers.
This assumption can be proved easily by showing that the expression for gravitational potential energy between a hollow sphere of mass (M) and a point mass (m) is the same as it would be for a pair of extended...

The Moon orbits around the Earth. In turn, the Earth (and other planets) orbit the Sun. The space directly above our atmosphere is filled with artificial satellites in orbit. One can examine the circular orbit, the simplest kind of orbit, to understand the relationship between the speed and the period of planets and satellites with respect to their positions and the bodies that they orbit.
Nicolaus Copernicus (1473-1543) first suggested that the Earth and all other planets orbit the Sun in...

Arc length represents the total distance traveled along a curve in space. For a moving object such as a helicopter, the path can be modeled by a vector-valued position function\begin{equation*}\mathbf{r}(t)=\langle x(t),y(t),z(t)\rangle\end{equation*}where t denotes time. Unlike displacement, which measures only the straight-line distance between two points, arc length accounts for every change in direction along the trajectory.To calculate arc length, the interval of motion is divided into...