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61-442: Lookouts have been traditionally placed in high on masts, in crow's nests and tops . The International Regulations for Preventing Collisions at Sea (1972) says in part: Lookouts report anything they see and or hear. When reporting contacts, lookouts give information such as, bearing of the object, which way the object is headed, target angles and position angles and what the contact is. Lookouts should be thoroughly familiar with

122-682: A circle instead of a line. The calculation takes every particle's x coordinate and maps it to an angle, θ i = x i x max 2 π {\displaystyle \theta _{i}={\frac {x_{i}}{x_{\max }}}2\pi } where x max is the system size in the x direction and x i ∈ [ 0 , x max ) {\displaystyle x_{i}\in [0,x_{\max })} . From this angle, two new points ( ξ i , ζ i ) {\displaystyle (\xi _{i},\zeta _{i})} can be generated, which can be weighted by

183-434: A good lookout point (hence the name) when built in an elevated position like a hillside or top of a hill. The term is sometimes used metaphorically for the topmost structures in buildings, towers , etc. Such structures are often referred to as a Widow's walk . Centre of mass In physics , the center of mass of a distribution of mass in space (sometimes referred to as the barycenter or balance point )

244-585: A supposed similarity to a kind of ritual wine goblet. According to William Scoresby Jr. , the crow's nest was invented in the 19th century by his father, William Scoresby Sr. , a whaler and also an Arctic explorer. However, Scoresby Sr. may simply have made an improvement on existing designs. Crow's nests appear in Egyptian reliefs as early as 1200 BC and on eighth to seventh century BC representations of Phoenician , Etruscan , and Boiotian ships. The crow’s nest disappears completely from depictions of ships after

305-413: 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. The linear and angular momentum of a collection of particles can be simplified by measuring the position and velocity of

366-422: 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. The center of mass plays an important role in astronomy and astrophysics, where it is commonly referred to as the barycenter . The barycenter

427-668: Is chosen as the center of mass these equations simplify to p = m v , L = ∑ i = 1 n m i ( r i − R ) × d d t ( r i − R ) + ∑ i = 1 n m i R × v {\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

488-468: 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 safer 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

549-432: 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

610-1708: Is the mass at the point r , g is the acceleration of gravity, and k ^ {\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, F = ∭ Q f ( r ) d V = ∭ Q ρ ( r ) d V ( − g k ^ ) = − M g k ^ , {\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 T = ∭ Q ( r − R ) × f ( r ) d V = ∭ Q ( r − R ) × ( − g ρ ( r ) d V k ^ ) = ( ∭ Q ρ ( r ) ( r − R ) d V ) × ( − g k ^ ) . {\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

671-503: 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

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732-903: Is the sum of the masses of all of the particles. These values are mapped back into a new angle, θ ¯ {\displaystyle {\overline {\theta }}} , from which the x coordinate of the center of mass can be obtained: θ ¯ = atan2 ⁡ ( − ζ ¯ , − ξ ¯ ) + π x com = x max θ ¯ 2 π {\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

793-474: 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

854-418: Is the unique point at any given time where the weighted relative position of the distributed mass sums to zero. For a rigid body containing its center of mass, 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

915-1282: Is the unit vector in the vertical direction). Let r 1 , r 2 , and r 3 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, T = ( r 1 − R ) × F 1 + ( r 2 − R ) × F 2 + ( r 3 − R ) × F 3 = 0 , {\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 R × ( − W k ^ ) = r 1 × F 1 + r 2 × F 2 + r 3 × F 3 . {\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

976-434: Is true for any internal forces that cancel in accordance with Newton's Third Law . The experimental determination of a body's center of mass makes use of gravity forces on the body and is based on the fact that the center of mass is the same as the center 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,

1037-418: 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 . 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

1098-416: Is very important to place the center of gravity at the center and well below the lift points. The center of mass of the adult human body is 10 cm above the trochanter (the femur joins the hip). 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:

1159-1141: The ( ξ , ζ ) {\displaystyle (\xi ,\zeta )} plane, these coordinates lie on a circle of radius 1. From the collection of ξ i {\displaystyle \xi _{i}} and ζ i {\displaystyle \zeta _{i}} values from all the particles, the averages ξ ¯ {\displaystyle {\overline {\xi }}} and ζ ¯ {\displaystyle {\overline {\zeta }}} are calculated. ξ ¯ = 1 M ∑ i = 1 n m i ξ i , ζ ¯ = 1 M ∑ i = 1 n m i ζ i , {\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

1220-515: The caboose , the cupola , was also called the crow's nest. It served for observation of the whole train when in motion. In hunting , a crow's nest is a blind-like structure where a hunter or a pair of hunters commit themselves to stalking game. A crow's nest is not a normal type of purchasable blind, but an improvised position, built by using locally discovered natural flora (tree branches, moss, snow (during winter) or sand (during summer), etc.). A crow's nest works in most environments and provides

1281-451: 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 (see Barycenter (astronomy) for details). 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 . The concept of center of gravity or weight

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1342-440: The percentage of the total mass divided between these two particles vary from 100% P 1 and 0% P 2 through 50% P 1 and 50% P 2 to 0% P 1 and 100% P 2 , then the center of mass R moves along the line from P 1 to P 2 . 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

1403-479: 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

1464-500: 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 ∑ i = 1 n m i ( r i − R ) = 0 . {\displaystyle \sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )=\mathbf {0} .} Solving this equation for R yields

1525-488: The center of mass is the same as the centroid of the volume. The coordinates R of the center of mass of a two-particle system, P 1 and P 2 , with masses m 1 and m 2 is given by R = m 1 r 1 + m 2 r 2 m 1 + m 2 . {\displaystyle \mathbf {R} ={{m_{1}\mathbf {r} _{1}+m_{2}\mathbf {r} _{2}} \over m_{1}+m_{2}}.} Let

1586-406: 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. An experimental method for locating the center of mass is to suspend

1647-493: 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

1708-502: 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. Knowing the location of the center of gravity when rigging

1769-460: The concept further. Newton's second law is reformulated with respect to the center of mass in Euler's first law . 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. In

1830-399: The coordinates R to obtain R = 1 M ∭ Q ρ ( r ) r d V , {\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 that ρ is constant, then

1891-623: The coordinates of the center of mass R * in the horizontal plane as, R ∗ = − 1 W k ^ × ( r 1 × F 1 + r 2 × F 2 + r 3 × F 3 ) . {\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

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1952-436: The distinction between the center-of-gravity and the mass-center. Any horizontal offset between the two will result in an applied torque. 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

2013-427: 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

2074-405: The formula R = ∑ i = 1 n m i r i ∑ i = 1 n m i . {\displaystyle \mathbf {R} ={\sum _{i=1}^{n}m_{i}\mathbf {r} _{i} \over \sum _{i=1}^{n}m_{i}}.} If the mass distribution is continuous with the density ρ( r ) within a solid Q , then

2135-439: 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 ∭ Q ρ ( r ) ( r − R ) d V = 0 . {\displaystyle \iiint _{Q}\rho (\mathbf {r} )\left(\mathbf {r} -\mathbf {R} \right)dV=\mathbf {0} .} Solve this equation for

2196-404: 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

2257-512: The mass of the particle x i {\displaystyle x_{i}} for the center of mass or given a value of 1 for the geometric center: ξ i = cos ⁡ ( θ i ) ζ i = sin ⁡ ( θ i ) {\displaystyle {\begin{aligned}\xi _{i}&=\cos(\theta _{i})\\\zeta _{i}&=\sin(\theta _{i})\end{aligned}}} In

2318-421: The name was coined because Scoresby's lookout platform resembled a crow's nest in a tree. Since the crow's nest is a point far away from the ship's center of mass , rotational movement of the ship is amplified and could lead to severe seasickness , even in accustomed sailors. Therefore, being sent to the crow's nest was also considered a punishment. In classic railroad trains , the box-like structure above

2379-463: 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

2440-520: The object. The center of mass will be the intersection of the two lines L 1 and L 2 obtained from the two experiments. 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

2501-741: 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, r i = ( r i − R ) + R , v i = d d t ( r i − R ) + v . {\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

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2562-415: 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

2623-461: 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. For particles in a system with periodic boundary conditions two particles can be neighbours even though they are on opposite sides of

2684-431: 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

2745-429: The reference point R is chosen so that it is the center of mass, then ∭ Q ρ ( r ) ( r − R ) d V = 0 , {\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

2806-401: 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 and further-from the planet (stronger and weaker gravity respectively) 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

2867-404: The seventh century, although the relationship between a raised vantage point and better visibility continued to be well understood. Theon of Smyrna wrote that by climbing the mast of a ship, one could see land that is invisible to those on deck. The first recorded appearance of the term was in 1807, used to describe Scoresby Sr.'s barrel crows nest platform. According to a popular naval legend,

2928-418: The space bounded by the four wheels even at angles far from the horizontal . 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

2989-1500: The system are p = d d t ( ∑ i = 1 n m i ( r i − R ) ) + ( ∑ i = 1 n m i ) v , {\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 L = ∑ i = 1 n m i ( r i − R ) × d d t ( r i − R ) + ( ∑ i = 1 n m i ) [ R × d d t ( r i − R ) + ( r i − R ) × v ] + ( ∑ i = 1 n m i ) R × v {\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

3050-615: 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, ( ξ ¯ , ζ ¯ ) = ( 0 , 0 ) {\displaystyle \left({\overline {\xi }},{\overline {\zeta }}\right)=(0,0)} , then θ ¯ {\displaystyle {\overline {\theta }}}

3111-440: 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

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3172-429: The term derives from the practice of Viking sailors, who carried crows or ravens in a cage secured to the top of the mast. In cases of poor visibility, a crow was released, and the navigator plotted a course corresponding to the bird's flight path because the bird invariably headed " as the crow flies " towards the nearest land. However, other naval scholars have found no evidence of the masthead crow cage and suggest

3233-471: 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

3294-421: The track in advance of an approaching train. This naval article is a stub . You can help Misplaced Pages by expanding it . This job-, occupation-, or vocation-related article is a stub . You can help Misplaced Pages by expanding it . Crow%27s nest A crow's nest is a structure in the upper part of the main mast of a ship or a structure that is used as a lookout point. On ships, this position ensured

3355-608: The various types of distress signals they may encounter at sea. By analogy, the term "lookout" is also used to describe a person who accompanies criminals during the commission of a crime , and warns them of the impending approach of hazards: that is, police or eyewitnesses . Although lookouts typically do not actually participate in the crime, they can nonetheless be charged with aiding and abetting or with conspiracy , or as accomplices . A lookout may be used when performing engineering works on an operational railway. They will be responsible for ensuring that all staff are cleared of

3416-442: The vertical line L , given by L ( t ) = R ∗ + t k ^ . {\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

3477-430: The volume. In a parallel gravity field the force f at each point r is given by, f ( r ) = − d m g k ^ = − ρ ( r ) d V g k ^ , {\displaystyle \mathbf {f} (\mathbf {r} )=-dm\,g\mathbf {\hat {k}} =-\rho (\mathbf {r} )\,dV\,g\mathbf {\hat {k}} ,} where dm

3538-407: 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

3599-450: The widest field of view for lookouts to spot approaching hazards, other ships, or land by using the naked eye or optical devices such as telescopes or binoculars . It should not be confused with the top , the platform in the upper part of each lower mast of a square-rigged sailing ship . The form of crow's nest used by the ancient Greeks and Romans was the carchesium ( Ancient Greek : καρχήσιον , karkhḗsion ), named after

3660-604: 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. 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, F 1 , F 2 , and F 3 that resist the weight of the object, W = − W k ^ {\displaystyle \mathbf {W} =-W\mathbf {\hat {k}} } ( k ^ {\displaystyle \mathbf {\hat {k}} }

3721-436: 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

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