[Ioan Tănăsescu (1875-1954)]

The perpendicular-axis theorem states that the moment of inertia of a planar object about an axis perpendicular to its plane is equal to the sum of the moments of inertia about two mutually perpendicular concurrent axes lying in the plane of the body.
Consider a circular disc of mass M and radius R lying along an x-y plane. The origin lies at the center of the disc, and the z-axis is perpendicular to the disc's plane. All three axes coincide at the disc's center. The moment of inertia of this...

Trigonometric identities are equations that relate trigonometric functions and hold for all angles within their domains. A fundamental identity among these is the Pythagorean identity, which arises directly from the geometry of the unit circle. For any angle θ, a point on the unit circle has coordinates (cos⁡ θ, sin ⁡θ), and since the radius of the circle is one, the Pythagorean Theorem gives:This identity serves as the basis for deriving additional identities. Dividing the Pythagorean identity...

The tonicity of a solution determines if a cell gains or loses water in that solution. The tonicity depends on the permeability of the cell membrane for different solutes and the concentration of nonpenetrating solutes in the solution within and outside of the cell. If a semipermeable membrane hinders the passage of some solutes but allows water to follow its concentration gradient, water moves from the side with low osmolarity (i.e., less solute) to the side with higher osmolarity (i.e.,...

Tonicity describes the amount of solute in a solution. The measure of the tonicity of a solution, or the total amount of solutes dissolved in a specific amount of solution, is called its osmolarity. Three terms—hypotonic, isotonic, and hypertonic—are used to relate the osmolarity of a cell to the osmolarity of the extracellular fluid that contains the cells. In a hypotonic solution, such as tap water, the extracellular fluid has a lower concentration of solutes than the fluid inside the cell,...

Changes in the environment of the early Earth drove the evolution of organisms. As prokaryotic organisms in the oceans began to photosynthesize, they produced oxygen. Eventually, oxygen saturated the oceans and entered the air, resulting in an increase in atmospheric oxygen concentration, known as the oxygen revolution approximately 2.3 billion years ago. Therefore, organisms that could use oxygen for cellular respiration had an advantage. More than 1.5 years ago, eukaryotic cells and...

The retina is a layer of nervous tissue at the back of the eye that transduces light into neural signals. This process, called phototransduction, is carried out by rod and cone photoreceptor cells in the back of the retina.