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50-639: The current official goal of the Marine Laboratory is to study marine environmental systems and conservation utilizing the resources of the facility's proximity to the ocean. It is a member of the National Association of Marine Laboratories and the Marine Sciences Education Consortium. The main campus of Duke University is not in close proximity to the ocean, but with the Marine Laboratory campus,

100-627: A unit vector u ^ {\displaystyle \mathbf {\hat {u}} } perpendicular to the plane of angular displacement, a scalar angular speed ω {\displaystyle \omega } results, where ω u ^ = ω , {\displaystyle \omega \mathbf {\hat {u}} ={\boldsymbol {\omega }},} and ω = v ⊥ r , {\displaystyle \omega ={\frac {v_{\perp }}{r}},} where v ⊥ {\displaystyle v_{\perp }}

150-409: A complex function of the configuration of the matter about the center of rotation and the orientation of the rotation for the various bits. For a rigid body , for instance a wheel or an asteroid, the orientation of rotation is simply the position of the rotation axis versus the matter of the body. It may or may not pass through the center of mass , or it may lie completely outside of the body. For

200-579: A fleet of research ships. Duke University shares Piver's Island with the National Oceanic and Atmospheric Administration (NOAA) , whichhas a 60,000 square foot marine laboratory. The current director of the National Laboratory is also an alum of Duke University. Duke has partnerships with other universities for marine research, such as Wittenberg University , Franklin and Marshall College , and Marquette University , for use of

250-469: A particular axis. However, if the particle's trajectory lies in a single plane , it is sufficient to discard the vector nature of angular momentum, and treat it as a scalar (more precisely, a pseudoscalar ). Angular momentum can be considered a rotational analog of linear momentum. Thus, where linear momentum p is proportional to mass m and linear speed v , p = m v , {\displaystyle p=mv,} angular momentum L

300-410: A particular interaction is called angular impulse , sometimes twirl . Angular impulse is the angular analog of (linear) impulse . The trivial case of the angular momentum L {\displaystyle L} of a body in an orbit is given by L = 2 π M f r 2 {\displaystyle L=2\pi Mfr^{2}} where M {\displaystyle M}

350-591: A renowned oceanographer and pioneer of Jacques Cousteau 's AquaLung Scuba device, received her M.S. and Ph.D. from Duke in 1956 and 1966, and has a connection to the Marine Laboratory. In 1990, the laboratory assumed much of the research of Fairleigh Dickinson University 's West Indies Laboratory for Underwater Research in St. Croix , Virgin Islands after Hurricane Hugo damaged the St. Croix facility. Cindy Lee Van Dover

400-430: Is also why hurricanes form spirals and neutron stars have high rotational rates. In general, conservation limits the possible motion of a system, but it does not uniquely determine it. The three-dimensional angular momentum for a point particle is classically represented as a pseudovector r × p , the cross product of the particle's position vector r (relative to some origin) and its momentum vector ;

450-405: Is always equal to the total torque on the system; the sum of all internal torques of any system is always 0 (this is the rotational analogue of Newton's third law of motion ). Therefore, for a closed system (where there is no net external torque), the total torque on the system must be 0, which means that the total angular momentum of the system is constant. The change in angular momentum for

500-659: Is always measured with respect to a fixed origin. Therefore, strictly speaking, L should be referred to as the angular momentum relative to that center . In the case of circular motion of a single particle, we can use I = r 2 m {\displaystyle I=r^{2}m} and ω = v / r {\displaystyle \omega ={v}/{r}} to expand angular momentum as L = r 2 m ⋅ v / r , {\displaystyle L=r^{2}m\cdot {v}/{r},} reducing to: L = r m v , {\displaystyle L=rmv,}

550-407: Is an important physical quantity because it is a conserved quantity  – the total angular momentum of a closed system remains constant. Angular momentum has both a direction and a magnitude, and both are conserved. Bicycles and motorcycles , flying discs , rifled bullets , and gyroscopes owe their useful properties to conservation of angular momentum. Conservation of angular momentum

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600-894: Is desired to know what effect the moving matter has on the point—can it exert energy upon it or perform work about it? Energy , the ability to do work , can be stored in matter by setting it in motion—a combination of its inertia and its displacement. Inertia is measured by its mass , and displacement by its velocity . Their product, ( amount of inertia ) × ( amount of displacement ) = amount of (inertia⋅displacement) mass × velocity = momentum m × v = p {\displaystyle {\begin{aligned}({\text{amount of inertia}})\times ({\text{amount of displacement}})&={\text{amount of (inertia⋅displacement)}}\\{\text{mass}}\times {\text{velocity}}&={\text{momentum}}\\m\times v&=p\\\end{aligned}}}

650-450: Is directed perpendicular to the plane of angular displacement, as indicated by the right-hand rule – so that the angular velocity is seen as counter-clockwise from the head of the vector. Conversely, the L {\displaystyle \mathbf {L} } vector defines the plane in which r {\displaystyle \mathbf {r} } and p {\displaystyle \mathbf {p} } lie. By defining

700-424: Is ignored in analysis of the system, except in regard to these interactions. This physics -related article is a stub . You can help Misplaced Pages by expanding it . This systems -related article is a stub . You can help Misplaced Pages by expanding it . Angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum ) is the rotational analog of linear momentum . It

750-401: Is known, the angular momentum L {\displaystyle L} is given by L = 16 15 π 2 ρ f r 5 {\displaystyle L={\frac {16}{15}}\pi ^{2}\rho fr^{5}} where ρ {\displaystyle \rho } is the sphere's density , f {\displaystyle f} is

800-445: Is proportional to moment of inertia I and angular speed ω measured in radians per second. L = I ω . {\displaystyle L=I\omega .} Unlike mass, which depends only on amount of matter, moment of inertia depends also on the position of the axis of rotation and the distribution of the matter. Unlike linear velocity, which does not depend upon the choice of origin, orbital angular velocity

850-401: Is related to the angular velocity of the rotation. Because moment of inertia is a crucial part of the spin angular momentum, the latter necessarily includes all of the complications of the former, which is calculated by multiplying elementary bits of the mass by the squares of their distances from the center of rotation. Therefore, the total moment of inertia, and the angular momentum, is

900-402: Is required to know the rate at which the position vector sweeps out angle, the direction perpendicular to the instantaneous plane of angular displacement, and the mass involved, as well as how this mass is distributed in space. By retaining this vector nature of angular momentum, the general nature of the equations is also retained, and can describe any sort of three-dimensional motion about

950-471: Is the angular momentum , sometimes called, as here, the moment of momentum of the particle versus that particular center point. The equation L = r m v {\displaystyle L=rmv} combines a moment (a mass m {\displaystyle m} turning moment arm r {\displaystyle r} ) with a linear (straight-line equivalent) speed v {\displaystyle v} . Linear speed referred to

1000-510: Is the cross product of the position vector r {\displaystyle \mathbf {r} } and the linear momentum p = m v {\displaystyle \mathbf {p} =m\mathbf {v} } of the particle. By the definition of the cross product, the L {\displaystyle \mathbf {L} } vector is perpendicular to both r {\displaystyle \mathbf {r} } and p {\displaystyle \mathbf {p} } . It

1050-509: Is the mass of the orbiting object, f {\displaystyle f} is the orbit's frequency and r {\displaystyle r} is the orbit's radius. The angular momentum L {\displaystyle L} of a uniform rigid sphere rotating around its axis, instead, is given by L = 4 5 π M f r 2 {\displaystyle L={\frac {4}{5}}\pi Mfr^{2}} where M {\displaystyle M}

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1100-401: Is the radius of gyration , the distance from the axis at which the entire mass m {\displaystyle m} may be considered as concentrated. Similarly, for a point mass m {\displaystyle m} the moment of inertia is defined as, I = r 2 m {\displaystyle I=r^{2}m} where r {\displaystyle r}

1150-483: Is the frequency of rotation and r {\displaystyle r} is the disk's radius. If instead the disk rotates about its diameter (e.g. coin toss), its angular momentum L {\displaystyle L} is given by L = 1 2 π M f r 2 {\displaystyle L={\frac {1}{2}}\pi Mfr^{2}} Just as for angular velocity , there are two special types of angular momentum of an object:

1200-438: Is the length of the moment arm , a line dropped perpendicularly from the origin onto the path of the particle. It is this definition, (length of moment arm) × (linear momentum) , to which the term moment of momentum refers. Another approach is to define angular momentum as the conjugate momentum (also called canonical momentum ) of the angular coordinate ϕ {\displaystyle \phi } expressed in

1250-407: Is the matter's momentum . Referring this momentum to a central point introduces a complication: the momentum is not applied to the point directly. For instance, a particle of matter at the outer edge of a wheel is, in effect, at the end of a lever of the same length as the wheel's radius, its momentum turning the lever about the center point. This imaginary lever is known as the moment arm . It has

1300-1002: Is the perpendicular component of the motion, as above. The two-dimensional scalar equations of the previous section can thus be given direction: L = I ω = I ω u ^ = ( r 2 m ) ω u ^ = r m v ⊥ u ^ = r ⊥ m v u ^ , {\displaystyle {\begin{aligned}\mathbf {L} &=I{\boldsymbol {\omega }}\\&=I\omega \mathbf {\hat {u}} \\&=\left(r^{2}m\right)\omega \mathbf {\hat {u}} \\&=rmv_{\perp }\mathbf {\hat {u}} \\&=r_{\perp }mv\mathbf {\hat {u}} ,\end{aligned}}} and L = r m v u ^ {\displaystyle \mathbf {L} =rmv\mathbf {\hat {u}} } for circular motion, where all of

1350-636: Is the perpendicular component of the motion. Expanding, L = r m v sin ⁡ ( θ ) , {\displaystyle L=rmv\sin(\theta ),} rearranging, L = r sin ⁡ ( θ ) m v , {\displaystyle L=r\sin(\theta )mv,} and reducing, angular momentum can also be expressed, L = r ⊥ m v , {\displaystyle L=r_{\perp }mv,} where r ⊥ = r sin ⁡ ( θ ) {\displaystyle r_{\perp }=r\sin(\theta )}

1400-561: Is the sphere's mass, f {\displaystyle f} is the frequency of rotation and r {\displaystyle r} is the sphere's radius. Thus, for example, the orbital angular momentum of the Earth with respect to the Sun is about 2.66 × 10 kg⋅m ⋅s , while its rotational angular momentum is about 7.05 × 10 kg⋅m ⋅s . In the case of a uniform rigid sphere rotating around its axis, if, instead of its mass, its density

1450-665: The Lagrangian of the mechanical system. Consider a mechanical system with a mass m {\displaystyle m} constrained to move in a circle of radius r {\displaystyle r} in the absence of any external force field. The kinetic energy of the system is T = 1 2 m r 2 ω 2 = 1 2 m r 2 ϕ ˙ 2 . {\displaystyle T={\tfrac {1}{2}}mr^{2}\omega ^{2}={\tfrac {1}{2}}mr^{2}{\dot {\phi }}^{2}.} And

1500-483: The spin angular momentum is the angular momentum about the object's centre of mass , while the orbital angular momentum is the angular momentum about a chosen center of rotation. The Earth has an orbital angular momentum by nature of revolving around the Sun , and a spin angular momentum by nature of its daily rotation around the polar axis. The total angular momentum is the sum of the spin and orbital angular momenta. In

1550-413: The surroundings or neighbourhood , and in thermodynamics , as the reservoir . Depending on the type of system, it may interact with the environment by exchanging mass , energy (including heat and work ), linear momentum , angular momentum , electric charge , or other conserved properties . In some disciplines, such as information theory , information may also be exchanged. The environment

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1600-479: The Marine Laboratory for students. The research originating from the laboratory has often been published in scientific journals, such as Policy Studies Journal , Ecology Letters , Marine Turtle , and Conservation Biology. Recently, the laboratory's research about the effect of plastic on sea coral has gained national media coverage. Other notable research includes the interaction of light pollution and marine life and studies of whale migration patterns. In 2017,

1650-414: The Marine Laboratory was featured in television series Xploration Station with Philippe Cousteau Jr. , the grandson of oceanographer Jacques Cousteau . Environmental systems In science and engineering , a system is the part of the universe that is being studied, while the environment is the remainder of the universe that lies outside the boundaries of the system. It is also known as

1700-456: The case of the Earth the primary conserved quantity is the total angular momentum of the solar system because angular momentum is exchanged to a small but important extent among the planets and the Sun. The orbital angular momentum vector of a point particle is always parallel and directly proportional to its orbital angular velocity vector ω , where the constant of proportionality depends on both

1750-1031: The center of rotation – circular , linear , or otherwise. In vector notation , the orbital angular momentum of a point particle in motion about the origin can be expressed as: L = I ω , {\displaystyle \mathbf {L} =I{\boldsymbol {\omega }},} where This can be expanded, reduced, and by the rules of vector algebra , rearranged: L = ( r 2 m ) ( r × v r 2 ) = m ( r × v ) = r × m v = r × p , {\displaystyle {\begin{aligned}\mathbf {L} &=\left(r^{2}m\right)\left({\frac {\mathbf {r} \times \mathbf {v} }{r^{2}}}\right)\\&=m\left(\mathbf {r} \times \mathbf {v} \right)\\&=\mathbf {r} \times m\mathbf {v} \\&=\mathbf {r} \times \mathbf {p} ,\end{aligned}}} which

1800-568: The central point is simply the product of the distance r {\displaystyle r} and the angular speed ω {\displaystyle \omega } versus the point: v = r ω , {\displaystyle v=r\omega ,} another moment. Hence, angular momentum contains a double moment: L = r m r ω . {\displaystyle L=rmr\omega .} Simplifying slightly, L = r 2 m ω , {\displaystyle L=r^{2}m\omega ,}

1850-528: The coordinate ϕ {\displaystyle \phi } is defined by p ϕ = ∂ L ∂ ϕ ˙ = m r 2 ϕ ˙ = I ω = L . {\displaystyle p_{\phi }={\frac {\partial {\mathcal {L}}}{\partial {\dot {\phi }}}}=mr^{2}{\dot {\phi }}=I\omega =L.} To completely define orbital angular momentum in three dimensions , it

1900-888: The effect of multiplying the momentum's effort in proportion to its length, an effect known as a moment . Hence, the particle's momentum referred to a particular point, ( moment arm ) × ( amount of inertia ) × ( amount of displacement ) = moment of (inertia⋅displacement) length × mass × velocity = moment of momentum r × m × v = L {\displaystyle {\begin{aligned}({\text{moment arm}})\times ({\text{amount of inertia}})\times ({\text{amount of displacement}})&={\text{moment of (inertia⋅displacement)}}\\{\text{length}}\times {\text{mass}}\times {\text{velocity}}&={\text{moment of momentum}}\\r\times m\times v&=L\\\end{aligned}}}

1950-440: The frequency of rotation and r {\displaystyle r} is the sphere's radius. In the simplest case of a spinning disk, the angular momentum L {\displaystyle L} is given by L = π M f r 2 {\displaystyle L=\pi Mfr^{2}} where M {\displaystyle M} is the disk's mass, f {\displaystyle f}

2000-460: The latter is p = m v in Newtonian mechanics . Unlike linear momentum, angular momentum depends on where this origin is chosen, since the particle's position is measured from it. Angular momentum is an extensive quantity ; that is, the total angular momentum of any composite system is the sum of the angular momenta of its constituent parts. For a continuous   rigid body or a fluid ,

2050-455: The mass of the particle and its distance from origin. The spin angular momentum vector of a rigid body is proportional but not always parallel to the spin angular velocity vector Ω , making the constant of proportionality a second-rank tensor rather than a scalar. Angular momentum is a vector quantity (more precisely, a pseudovector ) that represents the product of a body's rotational inertia and rotational velocity (in radians/sec) about

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2100-556: The motion is perpendicular to the radius r {\displaystyle r} . In the spherical coordinate system the angular momentum vector expresses as Angular momentum can be described as the rotational analog of linear momentum . Like linear momentum it involves elements of mass and displacement . Unlike linear momentum it also involves elements of position and shape . Many problems in physics involve matter in motion about some certain point in space, be it in actual rotation about it, or simply moving past it, where it

2150-574: The potential energy is U = 0. {\displaystyle U=0.} Then the Lagrangian is L ( ϕ , ϕ ˙ ) = T − U = 1 2 m r 2 ϕ ˙ 2 . {\displaystyle {\mathcal {L}}\left(\phi ,{\dot {\phi }}\right)=T-U={\tfrac {1}{2}}mr^{2}{\dot {\phi }}^{2}.} The generalized momentum "canonically conjugate to"

2200-658: The product of the radius of rotation r and the linear momentum of the particle p = m v {\displaystyle p=mv} , where v = r ω {\displaystyle v=r\omega } is the linear (tangential) speed . This simple analysis can also apply to non-circular motion if one uses the component of the motion perpendicular to the radius vector : L = r m v ⊥ , {\displaystyle L=rmv_{\perp },} where v ⊥ = v sin ⁡ ( θ ) {\displaystyle v_{\perp }=v\sin(\theta )}

2250-439: The quantity r 2 m {\displaystyle r^{2}m} is the particle's moment of inertia , sometimes called the second moment of mass. It is a measure of rotational inertia. The above analogy of the translational momentum and rotational momentum can be expressed in vector form: The direction of momentum is related to the direction of the velocity for linear movement. The direction of angular momentum

2300-535: The same body, angular momentum may take a different value for every possible axis about which rotation may take place. It reaches a minimum when the axis passes through the center of mass. For a collection of objects revolving about a center, for instance all of the bodies of the Solar System , the orientations may be somewhat organized, as is the Solar System, with most of the bodies' axes lying close to

2350-515: The system's axis. Their orientations may also be completely random. In brief, the more mass and the farther it is from the center of rotation (the longer the moment arm ), the greater the moment of inertia, and therefore the greater the angular momentum for a given angular velocity . In many cases the moment of inertia , and hence the angular momentum, can be simplified by, I = k 2 m , {\displaystyle I=k^{2}m,} where k {\displaystyle k}

2400-466: The total angular momentum is the volume integral of angular momentum density (angular momentum per unit volume in the limit as volume shrinks to zero) over the entire body. Similar to conservation of linear momentum, where it is conserved if there is no external force, angular momentum is conserved if there is no external torque . Torque can be defined as the rate of change of angular momentum, analogous to force . The net external torque on any system

2450-474: The university is able to conduct hands-on oceanographic studies. The site for the laboratory was selected in the early 1930s, with the first building completed by 1938. The original intention of the facility was to be a summer training facility and research facility for the university. By 1963, the facility had reached national recognition for its resources. At the time 75% of students and 40% of researchers came from other universities than Duke. Sylvia Earle ,

2500-550: Was named the first female-director of the center in 2007. She has initially worked at the laboratory in the 1970s. On June 5, 2017, the Marine Laboratory participated in a green-illumination protest with other buildings at Duke in support of the Paris Climate Accords along with similar actions at Harvard , Stanford , Yale , Columbia , and MIT . Since its inception in 1938, the campus has expanded significantly to include wet and dry laboratories as well as

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