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67-606: [REDACTED] Look up rolling  or thunder in Wiktionary, the free dictionary. Rolling Thunder may refer to: Arts [ edit ] Film [ edit ] Rolling Thunder (film) , a 1977 film starring William Devane Rolling Thunder (1996 film) , a film produced by Gary Adelson Rolling Thunder Pictures , a film distribution company Rolling Thunder Revue: A Bob Dylan Story by Martin Scorsese ,

134-429: A = F external m + I / r 2 {\displaystyle {\begin{aligned}F_{\text{net}}&={\frac {F_{\text{external}}}{1+{\frac {I}{mr^{2}}}}}={\frac {F_{\text{external}}}{1+\left({r_{\text{g}}}/{r}\right)^{2}}}\\[1ex]a&={\frac {F_{\text{external}}}{m+{I}/{r^{2}}}}\end{aligned}}} Because there is no slip, r α =

201-1231: A = F net m = ( F external 1 + ( r g / r ) 2 ) m = ( g m sin ⁡ ( θ ) 1 + ( r g / r ) 2 ) m = g m sin ⁡ ( θ ) m ( 1 + ( r g / r ) 2 ) = g sin ⁡ ( θ ) 1 + ( r g / r ) 2 {\displaystyle {\begin{aligned}a&={\frac {F_{\text{net}}}{m}}\\&={\frac {\left({\frac {F_{\text{external}}}{1+\left({r_{\text{g}}}/{r}\right)^{2}}}\right)}{m}}\\&={\frac {\left({\frac {gm\sin(\theta )}{1+\left({r_{\text{g}}}/{r}\right)^{2}}}\right)}{m}}\\&={\frac {gm\sin(\theta )}{m\left(1+\left({r_{\text{g}}}/{r}\right)^{2}\right)}}\\&={\frac {g\sin(\theta )}{1+\left({r_{\text{g}}}/{r}\right)^{2}}}\\\end{aligned}}} In

268-1669: A {\displaystyle r\alpha =a} holds. Substituting α {\displaystyle \alpha } and a {\displaystyle a} for the linear and rotational version of Newton's second law , then solving for F friction {\displaystyle F_{\text{friction}}} : r τ I = F net m r r F friction I = F net m F friction = I F net m r 2 {\displaystyle {\begin{aligned}r{\frac {\tau }{I}}&={\frac {F_{\text{net}}}{m}}\\r{\frac {rF_{\text{friction}}}{I}}&={\frac {F_{\text{net}}}{m}}\\F_{\text{friction}}&={\frac {IF_{\text{net}}}{mr^{2}}}\\\end{aligned}}} Expanding F friction {\displaystyle F_{\text{friction}}} in (1) : F net = F external − I F net m r 2 = F external − I m r 2 F net = F external 1 + I m r 2 {\displaystyle {\begin{aligned}F_{\text{net}}&=F_{\text{external}}-{\frac {IF_{\text{net}}}{mr^{2}}}\\&=F_{\text{external}}-{\frac {I}{mr^{2}}}F_{\text{net}}\\&={\frac {F_{\text{external}}}{1+{\frac {I}{mr^{2}}}}}\\\end{aligned}}} The last equality

335-425: A Lagrangian description of deformation. This approach greatly simplifies analysis by eliminating time-dependence, resulting in displacement, velocity, stress and strain fields that vary only spatially. Analysis procedures for finite element analysis of steady state rolling were first developed by Padovan , and are now featured in several commercial codes. Frame of reference In physics and astronomy ,

402-469: A coordinate system R with origin O . The corresponding set of axes, sharing the rigid body motion of the frame R {\displaystyle {\mathfrak {R}}} , can be considered to give a physical realization of R {\displaystyle {\mathfrak {R}}} . In a frame R {\displaystyle {\mathfrak {R}}} , coordinates are changed from R to R′ by carrying out, at each instant of time,

469-407: A coordinate system . If the basis vectors are orthogonal at every point, the coordinate system is an orthogonal coordinate system . An important aspect of a coordinate system is its metric tensor g ik , which determines the arc length ds in the coordinate system in terms of its coordinates: where repeated indices are summed over. As is apparent from these remarks, a coordinate system

536-404: A frame . According to this view, a frame is an observer plus a coordinate lattice constructed to be an orthonormal right-handed set of spacelike vectors perpendicular to a timelike vector. See Doran. This restricted view is not used here, and is not universally adopted even in discussions of relativity. In general relativity the use of general coordinate systems is common (see, for example,

603-403: A frame of reference (or reference frame ) is an abstract coordinate system , whose origin , orientation , and scale have been specified in physical space . It is based on a set of reference points , defined as geometric points whose position is identified both mathematically (with numerical coordinate values) and physically (signaled by conventional markers). An important special case

670-430: A physical frame of reference , a frame of reference , or simply a frame , is a physical concept related to an observer and the observer's state of motion. Here we adopt the view expressed by Kumar and Barve: an observational frame of reference is characterized only by its state of motion . However, there is lack of unanimity on this point. In special relativity, the distinction is sometimes made between an observer and

737-559: A professional wrestling attack Rolling Thunder Cyclocross Race , a cycling event Other uses [ edit ] Operation Rolling Thunder , a U.S. bombing campaign during the Vietnam War Rolling Thunder (organization) , a U.S. MIA/POW organization Rolling Thunder (person) , John Pope (1916–1997), a hippie spiritual leader Rolling Thunder Mountain , Wyoming, U.S. See also [ edit ] Thunder (disambiguation) Topics referred to by

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804-452: A (usually flat) object on a series of lined-up rollers, or wheels . The object on the wheels can be moved along them in a straight line, as long as the wheels are continuously replaced in the front (see history of bearings ). This method of primitive transportation is efficient when no other machinery is available. Today, the most practical application of objects on wheels are cars , trains , and other human transportation vehicles. Rolling

871-417: A 2019 pseudo-documentary film Music [ edit ] Rolling Thunder Revue , Bob Dylan's 1975–1976 musical tour The Bootleg Series Vol. 5: Bob Dylan Live 1975, The Rolling Thunder Revue , a live album recorded during the tour Rolling Thunder (album) , an album by Mickey Hart "Rolling Thunder" (march) , a march written by Henry Fillmore "Rolling Thunder", a song by A-ha from East of

938-516: A bowling ball. When static friction isn't enough, the friction becomes dynamic friction and slipping happens. The tangential force is opposite in direction to the external force, and therefore partially cancels it. The resulting net force and acceleration are: F net = F external 1 + I m r 2 = F external 1 + ( r g / r ) 2

1005-493: A definite state of motion at each event of spacetime. […] Within the context of special relativity and as long as we restrict ourselves to frames of reference in inertial motion, then little of importance depends on the difference between an inertial frame of reference and the inertial coordinate system it induces. This comfortable circumstance ceases immediately once we begin to consider frames of reference in nonuniform motion even within special relativity.…More recently, to negotiate

1072-511: A frame of reference in which the rolling plane is at rest (see animation), the instantaneous velocity of all the points of contact (for instance, a generating line segment of a cylinder) of the rolling object is zero. In practice, due to small deformations near the contact area, some sliding and energy dissipation occurs. Nevertheless, the resulting rolling resistance is much lower than sliding friction , and thus, rolling objects typically require much less energy to be moved than sliding ones. As

1139-400: A functional expansion like a Fourier series . In a physical problem, they could be spacetime coordinates or normal mode amplitudes. In a robot design , they could be angles of relative rotations, linear displacements, or deformations of joints . Here we will suppose these coordinates can be related to a Cartesian coordinate system by a set of functions: where x , y , z , etc. are

1206-480: A more restricted definition requires only that Newton's first law holds true; that is, a Newtonian inertial frame is one in which a free particle travels in a straight line at constant speed , or is at rest. These frames are related by Galilean transformations . These relativistic and Newtonian transformations are expressed in spaces of general dimension in terms of representations of the Poincaré group and of

1273-415: A reference plane. Most land vehicles use wheels and therefore rolling for displacement. Slip should be kept to a minimum (approximating pure rolling), otherwise loss of control and an accident may result. This may happen when the road is covered in snow, sand, or oil, when taking a turn at high speed or attempting to brake or accelerate suddenly. One of the most practical applications of rolling objects

1340-516: A result, such objects will more easily move, if they experience a force with a component along the surface, for instance gravity on a tilted surface, wind, pushing, pulling, or torque from an engine. Unlike cylindrical axially symmetric objects, the rolling motion of a cone is such that while rolling on a flat surface, its center of gravity performs a circular motion , rather than a linear motion . Rolling objects are not necessarily axially-symmetrical. Two well known non-axially-symmetrical rollers are

1407-404: A surface, an interface is formed through which normal and shear forces may be transmitted. For example, a tire contacting the road carries the weight (normal load) of the car as well as any shear forces arising due to acceleration, braking or steering. The deformations and motions in a steady rolling body can be efficiently characterized using an Eulerian description of rigid body rotation and

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1474-461: A truly inertial reference frame, which is one of free-fall.) A further aspect of a frame of reference is the role of the measurement apparatus (for example, clocks and rods) attached to the frame (see Norton quote above). This question is not addressed in this article, and is of particular interest in quantum mechanics , where the relation between observer and measurement is still under discussion (see measurement problem ). In physics experiments,

1541-445: Is a mathematical construct , part of an axiomatic system . There is no necessary connection between coordinate systems and physical motion (or any other aspect of reality). However, coordinate systems can include time as a coordinate, and can be used to describe motion. Thus, Lorentz transformations and Galilean transformations may be viewed as coordinate transformations . An observational frame of reference , often referred to as

1608-417: Is different from Wikidata All article disambiguation pages All disambiguation pages rolling Rolling where there is no sliding is referred to as pure rolling . By definition, there is no sliding when there is a frame of reference in which all points of contact on the rolling object have the same velocity as their counterparts on the surface on which the object rolls; in particular, for

1675-759: Is due to the fact rotational inertia of a point mass varies proportionally to the square of its distance to the axis. In the specific case of an object rolling in an inclined plane which experiences only static friction, normal force and its own weight, ( air drag is absent) the acceleration in the direction of rolling down the slope is: a = g sin ⁡ ( θ ) 1 + ( r g / r ) 2 {\displaystyle a={\frac {g\sin(\theta )}{1+\left({r_{\text{g}}}/{r}\right)^{2}}}} F external = g m sin ⁡ ( θ ) {\displaystyle F_{\text{external}}=gm\sin(\theta )}

1742-451: Is not inertial). In particle physics experiments, it is often useful to transform energies and momenta of particles from the lab frame where they are measured, to the center of momentum frame "COM frame" in which calculations are sometimes simplified, since potentially all kinetic energy still present in the COM frame may be used for making new particles. In this connection it may be noted that

1809-640: Is really quite different from that of a coordinate system. Frames differ just when they define different spaces (sets of rest points) or times (sets of simultaneous events). So the ideas of a space, a time, of rest and simultaneity, go inextricably together with that of frame. However, a mere shift of origin, or a purely spatial rotation of space coordinates results in a new coordinate system. So frames correspond at best to classes of coordinate systems. and from J. D. Norton: In traditional developments of special and general relativity it has been customary not to distinguish between two quite distinct ideas. The first

1876-488: Is taken beyond simple space-time coordinate systems by Brading and Castellani. Extension to coordinate systems using generalized coordinates underlies the Hamiltonian and Lagrangian formulations of quantum field theory , classical relativistic mechanics , and quantum gravity . We first introduce the notion of reference frame , itself related to the idea of observer : the reference frame is, in some sense,

1943-446: Is that of inertial reference frames , a stationary or uniformly moving frame. For n dimensions, n + 1 reference points are sufficient to fully define a reference frame. Using rectangular Cartesian coordinates , a reference frame may be defined with a reference point at the origin and a reference point at one unit distance along each of the n coordinate axes . In Einsteinian relativity , reference frames are used to specify

2010-432: Is the displacement between the particle and the rolling object's contact point (or line) with the surface, and ω is the angular velocity vector . Thus, despite that rolling is different from rotation around a fixed axis , the instantaneous velocity of all particles of the rolling object is the same as if it was rotating around an axis that passes through the point of contact with the same angular velocity. Any point in

2077-1037: Is the first formula for F net {\displaystyle F_{\text{net}}} ; using it together with Newton's second law, then reducing , the formula for a {\displaystyle a} is obtained: a = F net m = ( F external 1 + I m r 2 ) m = F external m ( 1 + I m r 2 ) = F external m + I / r 2 {\displaystyle {\begin{aligned}a&={\frac {F_{\text{net}}}{m}}\\&={\frac {\left({\frac {F_{\text{external}}}{1+{\frac {I}{mr^{2}}}}}\right)}{m}}\\&={\frac {F_{\text{external}}}{m\left(1+{\frac {I}{mr^{2}}}\right)}}\\&={\frac {F_{\text{external}}}{m+{I}/{r^{2}}}}\\\end{aligned}}} The radius of gyration can be incorporated in

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2144-437: Is the mass that would have a rotational inertia I {\displaystyle I} at distance r {\displaystyle r} from an axis of rotation. Therefore, the term I r 2 {\displaystyle {\tfrac {I}{r^{2}}}} may be thought of as the mass with linear inertia equivalent to the rolling object rotational inertia (around its center of mass). The action of

2211-413: Is the notion of a coordinate system, understood simply as the smooth, invertible assignment of four numbers to events in spacetime neighborhoods. The second, the frame of reference, refers to an idealized system used to assign such numbers […] To avoid unnecessary restrictions, we can divorce this arrangement from metrical notions. […] Of special importance for our purposes is that each frame of reference has

2278-411: Is the source of much confusion… the dependent functions such as velocity for example, are measured with respect to a physical reference frame, but one is free to choose any mathematical coordinate system in which the equations are specified. and this, also on the distinction between R {\displaystyle {\mathfrak {R}}} and [ R , R′ , etc. ]: The idea of a reference frame

2345-529: Is the use of rolling-element bearings , such as ball bearings , in rotating devices. Made of metal, the rolling elements are usually encased between two rings that can rotate independently of each other. In most mechanisms, the inner ring is attached to a stationary shaft (or axle). Thus, while the inner ring is stationary, the outer ring is free to move with very little friction . This is the basis for which almost all motors (such as those found in ceiling fans, cars, drills, etc.) rely on to operate. Alternatively,

2412-418: Is used to apply normal forces to a moving line of contact in various processes, for example in metalworking , printing , rubber manufacturing , painting . The simplest case of rolling is that of a rigid body rolling without slipping along a flat surface with its axis parallel to the surface (or equivalently: perpendicular to the surface normal ). The trajectory of any point is a trochoid ; in particular,

2479-419: Is what the physicist means as well. A coordinate system in mathematics is a facet of geometry or of algebra , in particular, a property of manifolds (for example, in physics, configuration spaces or phase spaces ). The coordinates of a point r in an n -dimensional space are simply an ordered set of n numbers: In a general Banach space , these numbers could be (for example) coefficients in

2546-658: The Galilean group . In contrast to the inertial frame, a non-inertial frame of reference is one in which fictitious forces must be invoked to explain observations. An example is an observational frame of reference centered at a point on the Earth's surface. This frame of reference orbits around the center of the Earth, which introduces the fictitious forces known as the Coriolis force , centrifugal force , and gravitational force . (All of these forces including gravity disappear in

2613-575: The Reuleaux triangle and the Meissner bodies . The oloid and the sphericon are members of a special family of developable rollers that develop their entire surface when rolling down a flat plane. Objects with corners, such as dice , roll by successive rotations about the edge or corner which is in contact with the surface. The construction of a specific surface allows even a perfect square wheel to roll with its centroid at constant height above

2680-483: The Schwarzschild solution for the gravitational field outside an isolated sphere ). There are two types of observational reference frame: inertial and non-inertial . An inertial frame of reference is defined as one in which all laws of physics take on their simplest form. In special relativity these frames are related by Lorentz transformations , which are parametrized by rapidity . In Newtonian mechanics,

2747-532: The n Cartesian coordinates of the point. Given these functions, coordinate surfaces are defined by the relations: The intersection of these surfaces define coordinate lines . At any selected point, tangents to the intersecting coordinate lines at that point define a set of basis vectors { e 1 , e 2 , ..., e n } at that point. That is: which can be normalized to be of unit length. For more detail see curvilinear coordinates . Coordinate surfaces, coordinate lines, and basis vectors are components of

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2814-539: The "Euclidean space carried by the observer". Let us give a more mathematical definition:… the reference frame is... the set of all points in the Euclidean space with the rigid body motion of the observer. The frame, denoted R {\displaystyle {\mathfrak {R}}} , is said to move with the observer.… The spatial positions of particles are labelled relative to a frame R {\displaystyle {\mathfrak {R}}} by establishing

2881-828: The Sun, West of the Moon "Rolling Thunder", a song by Conrad Sewell from Precious Other [ edit ] Rolling Thunder (journal) , an anarchist periodical Rolling Thunder (novel) , a novel by John Varley Rolling Thunder, a comics publishing company operated by Dave Dorman Sports, games and amusements [ edit ] Rolling Thunder (video game) , a side-scrolling action video game by Namco originally released in 1986 Rolling Thunder (roller coaster) , at Six Flags Great Adventure Rolling Thunder skate park , in London Rolling Thunder (Strongman) , an athletic event Rolling Thunder,

2948-1558: The axis of symmetry, then according to the parallel axis theorem, the rotational inertia associated with rolling is I rolling = m r 2 + I rotation {\displaystyle I_{\text{rolling}}=mr^{2}+I_{\text{rotation}}} (same as the rotational inertia of pure rotation around the point of contact). Using the general formula for kinetic energy of rotation, we have: K rolling = 1 2 I rolling ω 2 = 1 2 m r 2 ω 2 + 1 2 I rotation ω 2 = 1 2 m ( r ω ) 2 + 1 2 I rotation ω 2 = 1 2 m v c.o.m. 2 + 1 2 I rotation ω 2 = K translation + K rotation {\displaystyle {\begin{aligned}K_{\text{rolling}}&={\frac {1}{2}}I_{\text{rolling}}\omega ^{2}\\&={\frac {1}{2}}mr^{2}\omega ^{2}+{\frac {1}{2}}I_{\text{rotation}}\omega ^{2}\\&={\frac {1}{2}}m(r\omega )^{2}+{\frac {1}{2}}I_{\text{rotation}}\omega ^{2}\\&={\frac {1}{2}}mv_{\text{c.o.m.}}^{2}+{\frac {1}{2}}I_{\text{rotation}}\omega ^{2}\\&=K_{\text{translation}}+K_{\text{rotation}}\\\end{aligned}}} Differentiating

3015-595: The clocks and rods often used to describe observers' measurement equipment in thought, in practice are replaced by a much more complicated and indirect metrology that is connected to the nature of the vacuum , and uses atomic clocks that operate according to the standard model and that must be corrected for gravitational time dilation . (See second , meter and kilogram ). In fact, Einstein felt that clocks and rods were merely expedient measuring devices and they should be replaced by more fundamental entities based upon, for example, atoms and molecules. The discussion

3082-421: The external force results in a smaller net force by the dimensionless multiplicative factor 1 / ( 1 + I m r 2 ) {\displaystyle 1/\left(1+{\tfrac {I}{mr^{2}}}\right)} where I m r 2 {\displaystyle {\tfrac {I}{mr^{2}}}} represents the ratio of the aforesaid virtual mass to

3149-404: The external force upon an object in simple rotation may be conceptualized as accelerating the sum of the real mass and the virtual mass that represents the rotational inertia, which is m + I r 2 {\displaystyle m+{\tfrac {I}{r^{2}}}} . Since the work done by the external force is split between overcoming the translational and rotational inertia,

3216-836: The first formula for F net {\displaystyle F_{\text{net}}} as follows: r g = I m r g 2 = I m I m r 2 = ( I m ) r 2 = r g 2 r 2 = ( r g r ) 2 {\displaystyle {\begin{aligned}r_{\text{g}}&={\sqrt {\frac {I}{m}}}\\r_{\text{g}}^{2}&={\frac {I}{m}}\\{\frac {I}{mr^{2}}}&={\frac {\left({\frac {I}{m}}\right)}{r^{2}}}\\&={\frac {r_{\text{g}}^{2}}{r^{2}}}\\&=\left({\frac {r_{\text{g}}}{r}}\right)^{2}\\\end{aligned}}} Substituting

3283-412: The frame of reference in which the laboratory measurement devices are at rest is usually referred to as the laboratory frame or simply "lab frame." An example would be the frame in which the detectors for a particle accelerator are at rest. The lab frame in some experiments is an inertial frame, but it is not required to be (for example the laboratory on the surface of the Earth in many physics experiments

3350-408: The kinetic energy associated with simple rolling K rolling = K translation + K rotation {\displaystyle K_{\text{rolling}}=K_{\text{translation}}+K_{\text{rotation}}} Let r {\displaystyle r} be the distance between the center of mass and the point of contact; when the surface is flat, this is the radius of

3417-464: The last equality the denominator is the same as in the formula for force, but the factor m {\displaystyle m} disappears because its instance in the force of gravity cancels with its instance due to Newton's third law. r g / r {\displaystyle {r_{\text{g}}}/{r}} is specific to the object shape and mass distribution, it does not depend on scale or density. However, it will vary if

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3484-542: The latest equality above in the first formula for F net {\displaystyle F_{\text{net}}} the second formula for it: F net = F external 1 + ( r g / r ) 2 {\displaystyle F_{\text{net}}={\frac {F_{\text{external}}}{1+\left({r_{\text{g}}}/{r}\right)^{2}}}} I r 2 {\displaystyle {\tfrac {I}{r^{2}}}} has dimension of mass, and it

3551-415: The object actual mass and it is equal to ( r g / r ) 2 {\textstyle \left({r_{\text{g}}}/{r}\right)^{2}} where r g {\displaystyle r_{\text{g}}} is the radius of gyration corresponding to the object rotational inertia in pure rotation (not the rotational inertia in pure rolling). The square power

3618-468: The object around its widest cross section. Since the center of mass has an immediate velocity as if it was rotating around the point of contact, its velocity is v c.o.m. = r ω {\displaystyle v_{\text{c.o.m.}}=r\omega } . Due to symmetry, the object center of mass is a point in its axis. Let I rotation {\displaystyle I_{\text{rotation}}} be inertia of pure rotation around

3685-478: The object is made to roll with different radiuses; for instance, it varies between a train wheel set rolling normally (by its tire), and by its axle. It follows that given a reference rolling object, another object bigger or with different density will roll with the same acceleration. This behavior is the same as that of an object in free fall or an object sliding without friction (instead of rolling) down an inclined plane. When an axisymmetric deformable body contacts

3752-417: The object, both a net force and a torque are required. When external force with no torque acts on the rolling object‐surface system, there will be a tangential force at the point of contact between the surface and rolling object that provides the required torque as long as the motion is pure rolling; this force is usually static friction , for example, between the road and a wheel or between a bowling lane and

3819-451: The other hand, a coordinate system may be employed for many purposes where the state of motion is not the primary concern. For example, a coordinate system may be adopted to take advantage of the symmetry of a system. In a still broader perspective, the formulation of many problems in physics employs generalized coordinates , normal modes or eigenvectors , which are only indirectly related to space and time. It seems useful to divorce

3886-403: The outer ring may be attached to a fixed support bracket, allowing the inner ring to support an axle, allowing for rotational freedom of an axle . The amount of friction on the mechanism's parts depends on the quality of the ball bearings and how much lubrication is in the mechanism. Rolling objects are also frequently used as tools for transportation . One of the most basic ways is by placing

3953-599: The relation between linear and angular velocity , v c.o.m. = r ω {\displaystyle v_{\text{c.o.m.}}=r\omega } , with respect to time gives a formula relating linear and angular acceleration a = r α {\displaystyle a=r\alpha } . Applying Newton's second law : a = F net m = r α = r τ I . {\displaystyle a={\frac {F_{\text{net}}}{m}}=r\alpha ={\frac {r\tau }{I}}.} It follows that to accelerate

4020-626: The relationship between a moving observer and the phenomenon under observation. In this context, the term often becomes observational frame of reference (or observational reference frame ), which implies that the observer is at rest in the frame, although not necessarily located at its origin . A relativistic reference frame includes (or implies) the coordinate time , which does not equate across different reference frames moving relatively to each other. The situation thus differs from Galilean relativity , in which all possible coordinate times are essentially equivalent. The need to distinguish between

4087-430: The rolling object farther from the axis than the point of contact will temporarily move opposite to the direction of the overall motion when it is below the level of the rolling surface (for example, any point in the part of the flange of a train wheel that is below the rail). Since kinetic energy is entirely a function of an object mass and velocity, the above result may be used with the parallel axis theorem to obtain

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4154-467: The same coordinate transformation on the components of intrinsic objects (vectors and tensors) introduced to represent physical quantities in this frame . and this on the utility of separating the notions of R {\displaystyle {\mathfrak {R}}} and [ R , R′ , etc. ]: As noted by Brillouin, a distinction between mathematical sets of coordinates and physical frames of reference must be made. The ignorance of such distinction

4221-427: The same term [REDACTED] This disambiguation page lists articles associated with the title Rolling Thunder . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Rolling_Thunder&oldid=1184276778 " Category : Disambiguation pages Hidden categories: Short description

4288-484: The scale of their observations, as in macroscopic and microscopic frames of reference . In this article, the term observational frame of reference is used when emphasis is upon the state of motion rather than upon the coordinate choice or the character of the observations or observational apparatus. In this sense, an observational frame of reference allows study of the effect of motion upon an entire family of coordinate systems that could be attached to this frame. On

4355-395: The trajectory of any point in the object axis is a line, while the trajectory of any point in the object rim is a cycloid . The velocity of any point in the rolling object is given by v = ω × r {\displaystyle \mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {r} } , where r {\displaystyle \mathbf {r} }

4422-416: The various aspects of a reference frame for the discussion below. We therefore take observational frames of reference, coordinate systems, and observational equipment as independent concepts, separated as below: Although the term "coordinate system" is often used (particularly by physicists) in a nontechnical sense, the term "coordinate system" does have a precise meaning in mathematics, and sometimes that

4489-542: The various meanings of "frame of reference" has led to a variety of terms. For example, sometimes the type of coordinate system is attached as a modifier, as in Cartesian frame of reference . Sometimes the state of motion is emphasized, as in rotating frame of reference . Sometimes the way it transforms to frames considered as related is emphasized as in Galilean frame of reference . Sometimes frames are distinguished by

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