Center of mass

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Center_of_mass 

 

This toy uses the principles of center of mass to keep balance when sitting on a finger.

In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may be applied to cause a linear acceleration without an angular acceleration. Calculations in mechanics are often simplified when formulated with respect to the center of mass. It is a hypothetical point where the entire mass of an object may be assumed to be concentrated to visualise its motion. In other words, the center of mass is the particle equivalent of a given object for application of Newton's laws of motion.

In the case of a single rigid body, the center of mass is fixed in relation to the body, and if the body has uniform density, it will be located at the centroid. The center of mass may be located outside the physical body, as is sometimes the case for hollow or open-shaped objects, such as a horseshoe. In the case of a distribution of separate bodies, such as the planets of the Solar System, the center of mass may not correspond to the position of any individual member of the system.

The center of mass is a useful reference point for calculations in mechanics that involve masses distributed in space, such as the linear and angular momentum of planetary bodies and rigid body dynamics. In orbital mechanics, the equations of motion of planets are formulated as point masses located at the centers of mass. The center of mass frame is an inertial frame in which the center of mass of a system is at rest with respect to the origin of the coordinate system.

History

The concept of center of gravity or weight was studied extensively by the ancient Greek mathematician, physicist, and engineer Archimedes of Syracuse. He worked with simplified assumptions about gravity that amount to a uniform field, thus arriving at the mathematical properties of what we now call the center of mass. Archimedes showed that the torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass. In his work On Floating Bodies, Archimedes demonstrated that the orientation of a floating object is the one that makes its center of mass as low as possible. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes.

Other ancient mathematicians who contributed to the theory of the center of mass include Hero of Alexandria and Pappus of Alexandria. In the Renaissance and Early Modern periods, work by Guido Ubaldi, Francesco Maurolico, Federico Commandino, Evangelista Torricelli, Simon Stevin, Luca Valerio, Jean-Charles de la Faille, Paul Guldin, John Wallis, Christiaan Huygens, Louis Carré, Pierre Varignon, and Alexis Clairaut expanded the concept further.

Newton's second law is reformulated with respect to the center of mass in Euler's first law.

Definition

The center of mass is the unique point at the center of a distribution of mass in space that has the property that the weighted position vectors relative to this point sum to zero. In analogy to statistics, the center of mass is the mean location of a distribution of mass in space.

A system of particles

In the case of a system of particles P_i, i = 1, ..., n , each with mass m_i that are located in space with coordinates r_i, i = 1, ..., n , the coordinates R of the center of mass satisfy the condition

$${\displaystyle \sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )=\mathbf {0} .}$$

Solving this equation for R yields the formula

$${\displaystyle \mathbf {R} ={\frac {1}{M}}\sum _{i=1}^{n}m_{i}\mathbf {r} _{i},}$$

where ${\displaystyle M=\sum _{i=1}^{n}m_{i}}$ is the total mass of all of the particles.

A continuous volume

If the mass distribution is continuous with the density ρ(r) within a solid Q, then the integral of the weighted position coordinates of the points in this volume relative to the center of mass R over the volume V is zero, that is

$${\displaystyle \iiint _{Q}\rho (\mathbf {r} )\left(\mathbf {r} -\mathbf {R} \right)dV=0.}$$

Solve this equation for the coordinates R to obtain

$${\displaystyle \mathbf {R} ={\frac {1}{M}}\iiint _{Q}\rho (\mathbf {r} )\mathbf {r} \,dV,}$$

where M is the total mass in the volume.

If a continuous mass distribution has uniform density, which means ρ is constant, then the center of mass is the same as the centroid of the volume.

Barycentric coordinates

Further information: Barycentric coordinate system

The coordinates R of the center of mass of a two-particle system, P₁ and P₂, with masses m₁ and m₂ is given by

$${\displaystyle \mathbf {R} ={\frac {1}{m_{1}+m_{2}}}(m_{1}\mathbf {r} _{1}+m_{2}\mathbf {r} _{2}).}$$

Let the percentage of the total mass divided between these two particles vary from 100% P₁ and 0% P₂ through 50% P₁ and 50% P₂ to 0% P₁ and 100% P₂, then the center of mass R moves along the line from P₁ to P₂. The percentages of mass at each point can be viewed as projective coordinates of the point R on this line, and are termed barycentric coordinates. Another way of interpreting the process here is the mechanical balancing of moments about an arbitrary point. The numerator gives the total moment that is then balanced by an equivalent total force at the center of mass. This can be generalized to three points and four points to define projective coordinates in the plane, and in space, respectively.

Systems with periodic boundary conditions

For particles in a system with periodic boundary conditions two particles can be neighbours even though they are on opposite sides of the system. This occurs often in molecular dynamics simulations, for example, in which clusters form at random locations and sometimes neighbouring atoms cross the periodic boundary. When a cluster straddles the periodic boundary, a naive calculation of the center of mass will be incorrect. A generalized method for calculating the center of mass for periodic systems is to treat each coordinate, x and y and/or z, as if it were on a circle instead of a line. The calculation takes every particle's x coordinate and maps it to an angle,

$${\displaystyle \theta _{i}={\frac {x_{i}}{x_{\max }}}2\pi }$$

where x_max is the system size in the x direction and ${\displaystyle x_{i}\in [0,x_{\max })}$. From this angle, two new points ${\displaystyle (\xi _{i},\zeta _{i})}$ can be generated, which can be weighted by the mass of the particle ${\displaystyle x_{i}}$ for the center of mass or given a value of 1 for the geometric center:

$${\displaystyle {\begin{aligned}\xi _{i}&=\cos(\theta _{i})\\\zeta _{i}&=\sin(\theta _{i})\end{aligned}}}$$

In the ${\displaystyle (\xi ,\zeta )}$ plane, these coordinates lie on a circle of radius 1. From the collection of ${\displaystyle \xi _{i}}$ and ${\displaystyle \zeta _{i}}$ values from all the particles, the averages ${\displaystyle {\overline {\xi }}}$ and ${\displaystyle {\overline {\zeta }}}$ are calculated.

$${\displaystyle {\begin{aligned}{\overline {\xi }}&={\frac {1}{M}}\sum _{i=1}^{n}m_{i}\xi _{i},\\{\overline {\zeta }}&={\frac {1}{M}}\sum _{i=1}^{n}m_{i}\zeta _{i},\end{aligned}}}$$

where M is the sum of the masses of all of the particles.

These values are mapped back into a new angle, ${\displaystyle {\overline {\theta }}}$, from which the x coordinate of the center of mass can be obtained:

$${\displaystyle {\begin{aligned}{\overline {\theta }}&=\operatorname {atan2} \left(-{\overline {\zeta }},-{\overline {\xi }}\right)+\pi \\x_{\text{com}}&=x_{\max }{\frac {\overline {\theta }}{2\pi }}\end{aligned}}}$$

The process can be repeated for all dimensions of the system to determine the complete center of mass. The utility of the algorithm is that it allows the mathematics to determine where the "best" center of mass is, instead of guessing or using cluster analysis to "unfold" a cluster straddling the periodic boundaries. If both average values are zero, ${\displaystyle \left({\overline {\xi }},{\overline {\zeta }}\right)=(0,0)}$, then ${\displaystyle {\overline {\theta }}}$ is undefined. This is a correct result, because it only occurs when all particles are exactly evenly spaced. In that condition, their x coordinates are mathematically identical in a periodic system.

Center of gravity

"Center of gravity" redirects here. For other uses, see Center of gravity (disambiguation).

 

Main article: Centers of gravity in non-uniform fields

 

Diagram of an educational toy that balances on a point: the center of mass (C) settles below its support (P)

A body's center of gravity is the point around which the resultant torque due to gravity forces vanishes. Where a gravity field can be considered to be uniform, the mass-center and the center-of-gravity will be the same. However, for satellites in orbit around a planet, in the absence of other torques being applied to a satellite, the slight variation (gradient) in gravitational field between closer-to (stronger) and further-from (weaker) the planet can lead to a torque that will tend to align the satellite such that its long axis is vertical. In such a case, it is important to make the distinction between the center-of-gravity and the mass-center. Any horizontal offset between the two will result in an applied torque.

It is useful to note that the mass-center is a fixed property for a given rigid body (e.g. with no slosh or articulation), whereas the center-of-gravity may, in addition, depend upon its orientation in a non-uniform gravitational field. In the latter case, the center-of-gravity will always be located somewhat closer to the main attractive body as compared to the mass-center, and thus will change its position in the body of interest as its orientation is changed.

In the study of the dynamics of aircraft, vehicles and vessels, forces and moments need to be resolved relative to the mass center. That is true independent of whether gravity itself is a consideration. Referring to the mass-center as the center-of-gravity is something of a colloquialism, but it is in common usage and when gravity gradient effects are negligible, center-of-gravity and mass-center are the same and are used interchangeably.

In physics the benefits of using the center of mass to model a mass distribution can be seen by considering the resultant of the gravity forces on a continuous body. Consider a body Q of volume V with density ρ(r) at each point r in the volume. In a parallel gravity field the force f at each point r is given by,

$${\displaystyle \mathbf {f} (\mathbf {r} )=-dm\,g\mathbf {\hat {k}} =-\rho (\mathbf {r} )\,dV\,g\mathbf {\hat {k}} ,}$$

where dm is the mass at the point r, g is the acceleration of gravity, and ${\textstyle \mathbf {\hat {k}} }$ is a unit vector defining the vertical direction.

Choose a reference point R in the volume and compute the resultant force and torque at this point,

$${\displaystyle \mathbf {F} =\iiint _{Q}\mathbf {f} (\mathbf {r} )\,dV=\iiint _{Q}\rho (\mathbf {r} )\,dV\left(-g\mathbf {\hat {k}} \right)=-Mg\mathbf {\hat {k}} ,}$$

and

$${\displaystyle \mathbf {T} =\iiint _{Q}(\mathbf {r} -\mathbf {R} )\times \mathbf {f} (\mathbf {r} )\,dV=\iiint _{Q}(\mathbf {r} -\mathbf {R} )\times \left(-g\rho (\mathbf {r} )\,dV\,\mathbf {\hat {k}} \right)=\left(\iiint _{Q}\rho (\mathbf {r} )\left(\mathbf {r} -\mathbf {R} \right)dV\right)\times \left(-g\mathbf {\hat {k}} \right).}$$

If the reference point R is chosen so that it is the center of mass, then

$${\displaystyle \iiint _{Q}\rho (\mathbf {r} )\left(\mathbf {r} -\mathbf {R} \right)dV=0,}$$

which means the resultant torque T = 0. Because the resultant torque is zero the body will move as though it is a particle with its mass concentrated at the center of mass.

By selecting the center of gravity as the reference point for a rigid body, the gravity forces will not cause the body to rotate, which means the weight of the body can be considered to be concentrated at the center of mass.

Linear and angular momentum

The linear and angular momentum of a collection of particles can be simplified by measuring the position and velocity of the particles relative to the center of mass. Let the system of particles P_i, i = 1, ..., n of masses m_i be located at the coordinates r_i with velocities v_i. Select a reference point R and compute the relative position and velocity vectors,

$${\displaystyle \mathbf {r} _{i}=(\mathbf {r} _{i}-\mathbf {R} )+\mathbf {R} ,\quad \mathbf {v} _{i}={\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+\mathbf {v} .}$$

The total linear momentum and angular momentum of the system are

$${\displaystyle \mathbf {p} ={\frac {d}{dt}}\left(\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )\right)+\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {v} ,}$$

and

$${\displaystyle \mathbf {L} =\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )\times {\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+\left(\sum _{i=1}^{n}m_{i}\right)\left[\mathbf {R} \times {\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+(\mathbf {r} _{i}-\mathbf {R} )\times \mathbf {v} \right]+\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {R} \times \mathbf {v} }$$

If R is chosen as the center of mass these equations simplify to

$${\displaystyle \mathbf {p} =m\mathbf {v} ,\quad \mathbf {L} =\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )\times {\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+\sum _{i=1}^{n}m_{i}\mathbf {R} \times \mathbf {v} }$$

where m is the total mass of all the particles, p is the linear momentum, and L is the angular momentum.

The law of conservation of momentum predicts that for any system not subjected to external forces the momentum of the system will remain constant, which means the center of mass will move with constant velocity. This applies for all systems with classical internal forces, including magnetic fields, electric fields, chemical reactions, and so on. More formally, this is true for any internal forces that cancel in accordance with Newton's Third Law.

Locating the center of mass

Main article: Locating the center of mass

 

Plumb line method

The experimental determination of a body's centre of mass makes use of gravity forces on the body and is based on the fact that the centre of mass is the same as the centre of gravity in the parallel gravity field near the earth's surface.

The center of mass of a body with an axis of symmetry and constant density must lie on this axis. Thus, the center of mass of a circular cylinder of constant density has its center of mass on the axis of the cylinder. In the same way, the center of mass of a spherically symmetric body of constant density is at the center of the sphere. In general, for any symmetry of a body, its center of mass will be a fixed point of that symmetry.

In two dimensions

An experimental method for locating the center of mass is to suspend the object from two locations and to drop plumb lines from the suspension points. The intersection of the two lines is the center of mass.

The shape of an object might already be mathematically determined, but it may be too complex to use a known formula. In this case, one can subdivide the complex shape into simpler, more elementary shapes, whose centers of mass are easy to find. If the total mass and center of mass can be determined for each area, then the center of mass of the whole is the weighted average of the centers. This method can even work for objects with holes, which can be accounted for as negative masses.

A direct development of the planimeter known as an integraph, or integerometer, can be used to establish the position of the centroid or center of mass of an irregular two-dimensional shape. This method can be applied to a shape with an irregular, smooth or complex boundary where other methods are too difficult. It was regularly used by ship builders to compare with the required displacement and center of buoyancy of a ship, and ensure it would not capsize.

In three dimensions

An experimental method to locate the three-dimensional coordinates of the center of mass begins by supporting the object at three points and measuring the forces, F₁, F₂, and F₃ that resist the weight of the object, ${\displaystyle \mathbf {W} =-W\mathbf {\hat {k}} }$ (${\displaystyle \mathbf {\hat {k}} }$ is the unit vector in the vertical direction). Let r₁, r₂, and r₃ be the position coordinates of the support points, then the coordinates R of the center of mass satisfy the condition that the resultant torque is zero,

$${\displaystyle \mathbf {T} =(\mathbf {r} _{1}-\mathbf {R} )\times \mathbf {F} _{1}+(\mathbf {r} _{2}-\mathbf {R} )\times \mathbf {F} _{2}+(\mathbf {r} _{3}-\mathbf {R} )\times \mathbf {F} _{3}=0,}$$

or

$${\displaystyle \mathbf {R} \times \left(-W\mathbf {\hat {k}} \right)=\mathbf {r} _{1}\times \mathbf {F} _{1}+\mathbf {r} _{2}\times \mathbf {F} _{2}+\mathbf {r} _{3}\times \mathbf {F} _{3}.}$$

This equation yields the coordinates of the center of mass R* in the horizontal plane as,

$${\displaystyle \mathbf {R} ^{*}=-{\frac {1}{W}}\mathbf {\hat {k}} \times (\mathbf {r} _{1}\times \mathbf {F} _{1}+\mathbf {r} _{2}\times \mathbf {F} _{2}+\mathbf {r} _{3}\times \mathbf {F} _{3}).}$$

The center of mass lies on the vertical line L, given by

$${\displaystyle \mathbf {L} (t)=\mathbf {R} ^{*}+t\mathbf {\hat {k}} .}$$

The three-dimensional coordinates of the center of mass are determined by performing this experiment twice with the object positioned so that these forces are measured for two different horizontal planes through the object. The center of mass will be the intersection of the two lines L₁ and L₂ obtained from the two experiments.

Applications

Engineering designs

Automotive applications

Engineers try to design a sports car so that its center of mass is lowered to make the car handle better, which is to say, maintain traction while executing relatively sharp turns.

The characteristic low profile of the U.S. military Humvee was designed in part to allow it to tilt farther than taller vehicles without rolling over, by ensuring its low center of mass stays over the space bounded by the four wheels even at angles far from the horizontal.

Aeronautics

Main article: Center of gravity of an aircraft

The center of mass is an important point on an aircraft, which significantly affects the stability of the aircraft. To ensure the aircraft is stable enough to be safe to fly, the center of mass must fall within specified limits. If the center of mass is ahead of the forward limit, the aircraft will be less maneuverable, possibly to the point of being unable to rotate for takeoff or flare for landing. If the center of mass is behind the aft limit, the aircraft will be more maneuverable, but also less stable, and possibly unstable enough so as to be impossible to fly. The moment arm of the elevator will also be reduced, which makes it more difficult to recover from a stalled condition.

For helicopters in hover, the center of mass is always directly below the rotorhead. In forward flight, the center of mass will move forward to balance the negative pitch torque produced by applying cyclic control to propel the helicopter forward; consequently a cruising helicopter flies "nose-down" in level flight.

Astronomy

Main article: Barycenter

 

Two bodies orbiting their barycenter (red cross)

The center of mass plays an important role in astronomy and astrophysics, where it is commonly referred to as the barycenter. The barycenter is the point between two objects where they balance each other; it is the center of mass where two or more celestial bodies orbit each other. When a moon orbits a planet, or a planet orbits a star, both bodies are actually orbiting a point that lies away from the center of the primary (larger) body. For example, the Moon does not orbit the exact center of the Earth, but a point on a line between the center of the Earth and the Moon, approximately 1,710 km (1,062 miles) below the surface of the Earth, where their respective masses balance. This is the point about which the Earth and Moon orbit as they travel around the Sun. If the masses are more similar, e.g., Pluto and Charon, the barycenter will fall outside both bodies.

Rigging and safety

Knowing the location of the center of gravity when rigging is crucial, possibly resulting in severe injury or death if assumed incorrectly. A center of gravity that is at or above the lift point will most likely result in a tip-over incident. In general, the further the center of gravity below the pick point, the more safe the lift. There are other things to consider, such as shifting loads, strength of the load and mass, distance between pick points, and number of pick points. Specifically, when selecting lift points, it's very important to place the center of gravity at the center and well below the lift points.

Body motion

Main article: Kinesiology

In kinesiology and biomechanics, the center of mass is an important parameter that assists people in understanding their human locomotion. Typically, a human's center of mass is detected with one of two methods: the reaction board method is a static analysis that involves the person lying down on that instrument, and use of their static equilibrium equation to find their center of mass; the segmentation method relies on a mathematical solution based on the physical principle that the summation of the torques of individual body sections, relative to a specified axis, must equal the torque of the whole system that constitutes the body, measured relative to the same axis.

Human vestigiality

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Human_vestigiality 

 

The muscles connected to the ears of a human do not develop enough to have the same mobility allowed to monkeys. Arrows show the vestigial structure called Darwin's tubercle.

In the context of human evolution, human vestigiality involves those traits (such as organs or behaviors) occurring in humans that have lost all or most of their original function through evolution. Although structures called vestigial often appear functionless, a vestigial structure may retain lesser functions or develop minor new ones. In some cases, structures once identified as vestigial simply had an unrecognized function. Vestigial organs are sometimes called rudimentary organs.

The examples of human vestigiality are numerous, including the anatomical (such as the human tailbone, wisdom teeth, and inside corner of the eye), the behavioral (goose bumps and palmar grasp reflex), and molecular (pseudogenes). Many human characteristics are also vestigial in other primates and related animals.

History

Charles Darwin listed a number of putative human vestigial features, which he termed rudimentary, in The Descent of Man (1871). These included the muscles of the ear; wisdom teeth; the appendix; the tail bone; body hair; and the semilunar fold in the corner of the eye. Darwin also commented on the sporadic nature of many vestigial features, particularly musculature. Making reference to the work of the anatomist William Turner, Darwin highlighted a number of sporadic muscles which he identified as vestigial remnants of the panniculus carnosus, particularly the sternalis muscle.

In 1893, Robert Wiedersheim published The Structure of Man, a book on human anatomy and its relevance to man's evolutionary history. This book contained a list of 86 human organs that he considered vestigial, or as Wiedersheim himself explained: "Organs having become wholly or in part functionless, some appearing in the Embryo alone, others present during Life constantly or inconstantly. For the greater part Organs which may be rightly termed Vestigial." His list of supposedly vestigial organs included many of the examples on this page as well as others then mistakenly believed to be purely vestigial, such as the pineal gland, the thymus gland, and the pituitary gland. Some of these organs that had lost their obvious, original functions later turned out to have retained functions that had gone unrecognized before the discovery of hormones or many of the functions and tissues of the immune system. Examples included:

• the role of the pineal in the regulation of the circadian rhythm (neither the function nor even the existence of melatonin was yet known);
• discovery of the role of the thymus in the immune system lay many decades in the future; it remained a mystery organ until after the mid-20th century;
• the pituitary and hypothalamus with their many and varied hormones were far from understood, let alone the complexity of their interrelationships.

Historically, there was a trend not only to dismiss the vermiform appendix as being uselessly vestigial, but an anatomical hazard, a liability to dangerous inflammation. As late as the mid-20th century, many reputable authorities conceded it no beneficial function. This was a view supported, or perhaps inspired, by Darwin himself in the 1874 edition of his book The Descent of Man, and Selection in Relation to Sex. The organ's patent liability to appendicitis and its poorly understood role left the appendix open to blame for a number of possibly unrelated conditions. For example, in 1916, a surgeon claimed that removal of the appendix had cured several cases of trifacial neuralgia and other nerve pain about the head and face, even though he stated that the evidence for appendicitis in those patients was inconclusive. The discovery of hormones and hormonal principles, notably by Bayliss and Starling, argued against these views, but in the early twentieth century, there remained a great deal of fundamental research to be done on the functions of large parts of the digestive tract. In 1916, an author found it necessary to argue against the idea that the colon had no important function and that "the ultimate disappearance of the appendix is a coordinate action and not necessarily associated with such frequent inflammations as we are witnessing in the human".

There had been a long history of doubt about such dismissive views. Around 1920, the prominent surgeon Kenelm Hutchinson Digby documented previous observations, going back more than thirty years, that suggested lymphatic tissues, such as the tonsils and appendix, may have substantial immunological functions.

Anatomical

Appendix

Ileum, caecum and colon of rabbit, showing Appendix vermiformis on fully functional caecum

 

The human vermiform appendix on the vestigial caecum

In modern humans, the appendix is sometimes believed to be a vestige of a redundant organ that in ancestral species had digestive functions, much as it still does in extant species in which intestinal flora hydrolyze cellulose and similar indigestible plant materials. This view has changed over the past decades, with research suggesting that the appendix may serve an important purpose. In particular, it may serve as a reservoir for beneficial gut bacteria.

Some herbivorous animals, such as rabbits, have a terminal vermiform appendix and cecum that apparently bear patches of tissue with immune functions and may also be important in maintaining the composition of intestinal flora. It does not however seem to have much digestive function, if any, and is not present in all herbivores, even those with large caeca. As shown in the accompanying pictures however, the human appendix typically is about comparable to that of the rabbit's in size, though the caecum is reduced to a single bulge where the ileum empties into the colon. Some carnivorous animals may have appendices too, but seldom have more than vestigial caeca. In line with the possibility of vestigial organs developing new functions, some research suggests that the appendix may guard against the loss of symbiotic bacteria that aid in digestion, though that is unlikely to be a novel function, given the presence of vermiform appendices in many herbivores. Intestinal bacterial populations entrenched in the appendix may support quick re-establishment of the flora of the large intestine after an illness, poisoning, or after an antibiotic treatment depletes or otherwise causes harmful changes to the bacterial population of the colon.

A 2013 study, however, refutes the idea of an inverse relationship between cecum size and appendix size and presence. It is widely present in euarchontoglires (a superorder of mammals that includes rodents, lagomorphs and primates) and has also evolved independently in the diprotodont marsupials, monotremes, and is highly diverse in size and shape which could suggest it is not vestigial. Researchers deduce that the appendix has the ability to protect good bacteria in the gut. That way, when the gut is affected by a bout of diarrhea or other illness that cleans out the intestines, the good bacteria in the appendix can repopulate the digestive system and keep the person healthy.

Coccyx

The coccyx, or tailbone, is the remnant of a lost tail. All mammals have a tail at some point in their development; in humans, it is present for a period of 4 weeks, during stages 14 to 22 of human embryogenesis. This tail is most prominent in human embryos 31–35 days old. The tailbone, located at the end of the spine, has lost its original function in assisting balance and mobility, though it still serves some secondary functions, such as being an attachment point for muscles, which explains why it has not degraded further. The coccyx serves as an attachment site for tendons, ligaments, and muscles. It also functions as an insertion point of some of the muscles of the pelvic floor. In rare cases, congenital defect results in a short tail-like structure being present at birth. Twenty-three cases of human babies born with such a structure have been reported in the medical literature since 1884. In rare cases such as these, the spine and skull were determined to be entirely normal. The only abnormality was that of a tail approximately twelve centimeters long. These tails, though of no deleterious effect, were almost always surgically removed.

Wisdom teeth

Wisdom teeth are vestigial third molars that human ancestors used to help in grinding down plant tissue. The common postulation is that the skulls of human ancestors had larger jaws with more teeth, which were possibly used to help chew down foliage to compensate for a lack of ability to efficiently digest the cellulose that makes up a plant cell wall. As human diets changed, smaller jaws were naturally selected, yet the third molars, or "wisdom teeth", still commonly develop in human mouths. In modern human populations, wisdom teeth have become useless and often present harmful complications to the extent that surgical procedures are frequently performed to remove them.

Agenesis (failure to develop) of wisdom teeth in human populations ranges from zero in Tasmanian Aboriginals to nearly 100% in indigenous Mexicans. The difference is related to the PAX9 gene (and perhaps other genes).

Vomeronasal organ

In some animals, the vomeronasal organ (VNO) is part of a second, completely separate sense of smell, known as the accessory olfactory system. Many studies have been performed to find if there is an actual presence of a VNO in adult human beings. Trotier et al. estimated that around 92% of their subjects who had not had septal surgery had at least one intact VNO. Kjaer and Fisher Hansen, on the other hand, stated that the VNO structure disappeared during fetal development as it does for some primates. However, Smith and Bhatnagar (2000) asserted that Kjaer and Fisher Hansen simply missed the structure in older fetuses. Won (2000) found evidence of a VNO in 13 of his 22 cadavers (59.1%) and in 22 of his 78 living patients (28.2%). Given these findings, some scientists have argued that there is a VNO in adult human beings. However, most investigators have sought to identify the opening of the vomeronasal organ in humans, rather than identify the tubular epithelial structure itself. Thus it has been argued that such studies, employing macroscopic observational methods, have sometimes missed or even misidentified the vomeronasal organ.

Among studies that use microanatomical methods, there is no reported evidence that human beings have active sensory neurons like those in working vomeronasal systems of other animals. Furthermore, there is no evidence to date that suggests there are nerve and axon connections between any existing sensory receptor cells that may be in the adult human VNO and the brain. Likewise, there is no evidence for any accessory olfactory bulb in adult human beings, and the key genes involved in VNO function in other mammals have become pseudogenes in human beings. Therefore, while the presence of a structure in adult human beings is debated, a review of the scientific literature by Tristram Wyatt concluded, "most in the field ... are sceptical about the likelihood of a functional VNO in adult human beings on current evidence."

Ear

Top: Muscles of the human ear.
Bottom: The non-vestigial auricular muscle in the donkey can help it to move its ears like antennae.

The ears of a macaque monkey and most other monkeys have far more developed muscles than those of humans, and therefore have the capability to move their ears to better hear potential threats. Humans and other primates such as the orangutan and chimpanzee however have ear muscles that are minimally developed and non-functional, yet still large enough to be identifiable. A muscle attached to the ear that cannot move the ear, for whatever reason, can no longer be said to have any biological function. In humans there is variability in these muscles, such that some people are able to move their ears in various directions, and it can be possible for others to gain such movement by repeated trials. In such primates, the inability to move the ear is compensated mainly by the ability to turn the head on a horizontal plane, an ability which is not common to most monkeys—a function once provided by one structure is now replaced by another.

The outer structure of the ear also shows some vestigial features, such as the node or point on the helix of the ear known as Darwin's tubercle which is found in around 10% of the population.

Eye

The plica semilunaris is a small fold of tissue on the inside corner of the eye. It is the vestigial remnant of the nictitating membrane, i.e., third eyelid, an organ that is fully functional in some other species of mammals. Its associated muscles are also vestigial. Only one species of primate, the Calabar angwantibo, is known to have a functioning nictitating membrane.

The orbitalis muscle is a vestigial or rudimentary nonstriated muscle (smooth muscle) of the eye that crosses from the infraorbital groove and sphenomaxillary fissure and is intimately united with the periosteum of the orbit. It was described by Johannes Peter Müller and is often called Müller's muscle. The muscle forms an important part of the lateral orbital wall in some animals, but in humans it is not known to have any significant function.

Reproductive system

Main article: List of homologues of the human reproductive system

Genitalia

In the internal genitalia of each human sex, there are some residual organs of mesonephric and paramesonephric ducts during embryonic development:

• Gartner's duct
• Epoophoron
• Vesicular appendages of epoophoron
• Paroophoron

Human vestigial structures also include leftover embryological remnants that once served a function during development, such as the belly button, and analogous structures between biological sexes. For example, men are also born with two nipples, which are not known to serve a function compared to women. In regards to genitourinary development, both internal and external genitalia of male and female fetuses have the ability to fully or partially form their analogous phenotype of the opposite biological sex if exposed to a lack/overabundance of androgens or the SRY gene during fetal development. Examples of vestigial remnants of genitourinary development include the hymen, which is a membrane that surrounds or partially covers the external vaginal opening that derives from the sinus tubercle during fetal development and is homologous to the male seminal colliculus. Some researchers have hypothesized that the persistence of the hymen may be to provide temporary protection from infection, as it separates the vaginal lumen from the urogenital sinus cavity during development. Other examples include the glans penis and the clitoris, the labia minora and the ventral penis, and the ovarian follicles and the seminiferous tubules.

In modern times, there is controversy regarding whether the foreskin is a vital or vestigial structure. In 1949, British physician Douglas Gairdner noted that the foreskin plays an important protective role in newborns. He wrote, "It is often stated that the prepuce is a vestigial structure devoid of function ... However, it seems to be no accident that during the years when the child is incontinent the glans is completely clothed by the prepuce, for, deprived of this protection, the glans becomes susceptible to injury from contact with sodden clothes or napkin." During the physical act of sex, the foreskin reduces friction, which can reduce the need for additional sources of lubrication. "Some medical researchers, however, claim circumcised men enjoy sex just fine and that, in view of recent research on HIV transmission, the foreskin causes more trouble than it's worth." The area of the outer foreskin measures between 7 and 100 cm², and the inner foreskin measures between 18 and 68 cm², which is a wide range. Regarding vestigial structures, Charles Darwin wrote, "An organ, when rendered useless, may well be variable, for its variations cannot be checked by natural selection." Charles Darwin speculated that the sensitivity of the foreskin to fine touch might have served as an "early warning system" in our naked ancestors while it protected the glans from the intrusion of biting insects and parasites.

Musculature

A number of muscles in the human body are thought to be vestigial, either by virtue of being greatly reduced in size compared to homologous muscles in other species, by having become principally tendonous, or by being highly variable in their frequency within or between populations.

Head

The occipitalis minor is a muscle in the back of the head which normally joins to the auricular muscles of the ear. This muscle is very sporadic in frequency—always present in Malays, present in 56% of Africans, 50% of Japanese, and 36% of Europeans, and nonexistent in the Khoikhoi people of southwestern Africa and in Melanesians. Other small muscles in the head associated with the occipital region and the post-auricular muscle complex are often variable in their frequency.

The platysma, a quadrangular (four sides) muscle in a sheet-like configuration, is a vestigial remnant of the panniculous carnosus of animals. In horses, it is the muscle that allows it to flick a fly off its back.

Face

In many lower animals, the upper lip and sinus area is associated with whiskers or vibrissae which serve a sensory function. In humans, these whiskers do not exist but there are still sporadic cases where elements of the associated vibrissal capsular muscles or sinus hair muscles can be found. Based on histological studies of the upper lips of 20 cadavers, Tamatsu et al. found that structures resembling such muscles were present in 35% (7/20) of their specimens.

Arm

The palmaris longus muscle is seen as a small tendon between the flexor carpi radialis and the flexor carpi ulnaris, although it is not always present. The muscle is absent in about 14% of the population, however this varies greatly with ethnicity. It is believed that this muscle actively participated in the arboreal locomotion of primates, but currently has no function, because it does not provide more grip strength. One study has shown the prevalence of palmaris longus agenesis in 500 Indian patients to be 17.2% (8% bilateral and 9.2% unilateral). The palmaris is a popular source of tendon material for grafts and this has prompted studies which have shown the absence of the palmaris does not have any appreciable effect on grip strength.

The levator claviculae muscle in the posterior triangle of the neck is a supernumerary muscle present in only 2–3% of all people but nearly always present in most mammalian species, including gibbons and orangutans.

Torso

The pyramidalis muscle of the abdomen is a small and triangular muscle, anterior to the rectus abdominis, and contained in the rectus sheath. It is absent in 20% of humans and when absent, the lower end of the rectus then becomes proportionately increased in size. Anatomical studies suggest that the forces generated by the pyramidalis muscles are relatively small.

The latissimus dorsi muscle of the back has several sporadic variations. One particular variant is the existence of the dorsoepitrochlearis or latissimocondyloideus muscle which is a muscle passing from the tendon of the latissimus dorsi to the long head of the triceps brachii. It is notable due to its well developed character in other apes and monkeys, where it is an important climbing muscle, namely the dorsoepitrochlearis brachii. This muscle is found in ≈5% of humans.

Leg

The plantaris muscle is composed of a thin muscle belly and a long thin tendon. The muscle belly is approximately 5–10 centimetres (2–4 inches) long, and is absent in 7–10% of the human population. It has some weak functionality in moving the knee and ankle but is generally considered redundant and is often used as a source of tendon for grafts. The long, thin tendon of the plantaris is humorously called "the freshman's nerve", as it is often mistaken for a nerve by new medical students.

Tongue

Another example of human vestigiality occurs in the tongue, specifically the chondroglossus muscle. In a morphological study of 100 Japanese cadavers, it was found that 86% of fibers identified were solid and bundled in the appropriate way to facilitate speech and mastication. The other 14% of fibers were short, thin and sparse – nearly useless, and thus concluded to be of vestigial origin.

Breasts

Extra nipples or breasts sometimes appear along the mammary lines of humans, appearing as a remnant of mammalian ancestors who possessed more than two nipples or breasts. One recent report demonstrated that all healthy young men and women who participated in an anatomic study of the front surface of the body exhibited 8 pairs of focal fat mounds running along the embryological mammary ridges from axillae to the upper inner thighs. These were always located in the same relative anatomic sites – analogous to the loci of breasts in other placental mammals – and often had nipple-like moles or extra hairs located atop the mounds. Therefore, focal fatty prominences on the fronts of human torsos likely represent chains of vestigial breasts composed of primordial breast fat.

Behavioral

See also: Vestigial response § In humans

Humans also bear some vestigial behaviors and reflexes.

Goose bumps

Goose bumps are an example of a vestigial human reaction to stress.

The formation of goose bumps in humans under stress is a vestigial reflex; a possible function in the distant evolutionary ancestors of humanity was to raise the body's hair, making the ancestor appear larger and scaring off predators. Raising the hair is also used to trap an extra layer of air, keeping an animal warm. Due to the diminished amount of hair in humans, the reflex formation of goose bumps when cold is also vestigial.

Palmar grasp reflex

The palmar grasp reflex is thought to be a vestigial behavior in human infants. When placing a finger or object to the palm of an infant, it will securely grasp it. This grasp is found to be rather strong. Some infants—37% according to a 1932 study—are able to support their own weight from a rod, although there is no way they can cling to their mother. The grasp is also evident in the feet too. When a baby is sitting down, its prehensile feet assume a curled-in posture, similar to that observed in an adult chimp. An ancestral primate would have had sufficient body hair to which an infant could cling, unlike modern humans, thus allowing its mother to escape from danger, such as climbing up a tree in the presence of a predator without having to occupy her hands holding her baby.

Hiccup

It has been proposed that the hiccup is an evolutionary remnant of earlier amphibian respiration. Amphibians such as tadpoles gulp air and water across their gills via a rather simple motor reflex akin to mammalian hiccuping. The motor pathways that enable hiccuping form early during fetal development, before the motor pathways that enable normal lung ventilation form. Thus, according to recapitulation theory, the hiccup is evolutionarily antecedent to modern lung respiration. Additionally, they point out that hiccups and amphibian gulping are inhibited by elevated CO₂ and may be stopped by GABA_B receptor agonists, illustrating a possible shared physiology and evolutionary heritage. These proposals may explain why premature infants spend 2.5% of their time hiccuping, possibly gulping like amphibians, as their lungs are not yet fully formed. Fetal intrauterine hiccups are of two types. The physiological type occurs before 28 weeks after conception and tend to last five to ten minutes. These hiccups are part of fetal development and are associated with the myelination of the phrenic nerve, which primarily controls the thoracic diaphragm. The phylogeny hypothesis explains how the hiccup reflex might have evolved, and if there is not an explanation, it may explain hiccups as an evolutionary remnant, held-over from our amphibious ancestors. This hypothesis has been questioned because of the existence of the afferent loop of the reflex, the fact that it does not explain the reason for glottic closure, and because the very short contraction of the hiccup is unlikely to have a significant strengthening effect on the slow-twitch muscles of respiration.

Molecular

Further information: Pseudogene

There are also vestigial molecular structures in humans, which are no longer in use but may indicate common ancestry with other species. One example of this is L-gulonolactone oxidase, a gene that is functional in most other mammals and produces an enzyme that synthesizes vitamin C. In humans and other members of the suborder Haplorrhini, a mutation disabled the gene and made it unable to produce the enzyme. However, the remains of the gene are still present in the human genome as a vestigial genetic sequence called a pseudogene.