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97-408: 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 a frame of reference in which the rolling plane is at rest (see animation), the instantaneous velocity of all

194-429: 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 =

291-1231: 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

388-1669: 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

485-446: A n ) {\displaystyle \mathbb {n} =\left(a_{1},\ldots ,a_{n}\right)} is a normal. The definition of a normal to a surface in three-dimensional space can be extended to ( n − 1 ) {\displaystyle (n-1)} -dimensional hypersurfaces in R n . {\displaystyle \mathbb {R} ^{n}.} A hypersurface may be locally defined implicitly as

582-491: A , 0 , 0 ) , {\displaystyle (a,0,0),} where a ≠ 0 , {\displaystyle a\neq 0,} the rows of the Jacobian matrix are ( 0 , 0 , 1 ) {\displaystyle (0,0,1)} and ( 0 , a , 0 ) . {\displaystyle (0,a,0).} Thus the normal affine space is the plane of equation x =

679-417: A . {\displaystyle x=a.} Similarly, if b ≠ 0 , {\displaystyle b\neq 0,} the normal plane at ( 0 , b , 0 ) {\displaystyle (0,b,0)} is the plane of equation y = b . {\displaystyle y=b.} At the point ( 0 , 0 , 0 ) {\displaystyle (0,0,0)}

776-424: 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 ,

873-442: A convex polygon (such as a triangle ), a surface normal can be calculated as the vector cross product of two (non-parallel) edges of the polygon. For a plane given by the general form plane equation a x + b y + c z + d = 0 , {\displaystyle ax+by+cz+d=0,} the vector n = ( a , b , c ) {\displaystyle \mathbf {n} =(a,b,c)}

970-469: 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,

1067-407: 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

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1164-404: 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,

1261-403: 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

1358-430: 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

1455-452: 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

1552-516: 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

1649-595: A curve or to a surface is the Euclidean distance between Q and its foot P . The normal direction to a space curve is: where R = κ − 1 {\displaystyle R=\kappa ^{-1}} is the radius of curvature (reciprocal curvature ); T {\displaystyle \mathbf {T} } is the tangent vector , in terms of the curve position r {\displaystyle \mathbf {r} } and arc-length s {\displaystyle s} : For

1746-493: 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

1843-400: 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

1940-403: A manifold at point P {\displaystyle P} is the set of vectors which are orthogonal to the tangent space at P . {\displaystyle P.} Normal vectors are of special interest in the case of smooth curves and smooth surfaces . The normal is often used in 3D computer graphics (notice the singular, as only one normal will be defined) to determine

2037-480: 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

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2134-412: A normal to S is by definition a normal to a tangent plane, given by the cross product of the partial derivatives n = ∂ r ∂ s × ∂ r ∂ t . {\displaystyle \mathbf {n} ={\frac {\partial \mathbf {r} }{\partial s}}\times {\frac {\partial \mathbf {r} }{\partial t}}.} If

2231-406: A point P , {\displaystyle P,} the normal vector space is the vector space generated by the values at P {\displaystyle P} of the gradient vectors of the f i . {\displaystyle f_{i}.} In other words, a variety is defined as the intersection of k {\displaystyle k} hypersurfaces, and

2328-415: 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

2425-516: 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

2522-408: A surface S {\displaystyle S} is given implicitly as the set of points ( x , y , z ) {\displaystyle (x,y,z)} satisfying F ( x , y , z ) = 0 , {\displaystyle F(x,y,z)=0,} then a normal at a point ( x , y , z ) {\displaystyle (x,y,z)} on

2619-415: A surface does not have a tangent plane at a singular point , it has no well-defined normal at that point: for example, the vertex of a cone . In general, it is possible to define a normal almost everywhere for a surface that is Lipschitz continuous . The normal to a (hyper)surface is usually scaled to have unit length , but it does not have a unique direction, since its opposite is also a unit normal. For

2716-417: A surface which is the topological boundary of a set in three dimensions, one can distinguish between two normal orientations , the inward-pointing normal and outer-pointing normal . For an oriented surface , the normal is usually determined by the right-hand rule or its analog in higher dimensions. If the normal is constructed as the cross product of tangent vectors (as described in the text above), it

2813-405: A surface's orientation toward a light source for flat shading , or the orientation of each of the surface's corners ( vertices ) to mimic a curved surface with Phong shading . The foot of a normal at a point of interest Q (analogous to the foot of a perpendicular ) can be defined at the point P on the surface where the normal vector contains Q . The normal distance of a point Q to

2910-404: 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

3007-461: 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,

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3104-418: Is parameterized by a system of curvilinear coordinates r ( s , t ) = ( x ( s , t ) , y ( s , t ) , z ( s , t ) ) , {\displaystyle \mathbf {r} (s,t)=(x(s,t),y(s,t),z(s,t)),} with s {\displaystyle s} and t {\displaystyle t} real variables, then

3201-445: 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

3298-460: Is a pseudovector . When applying a transform to a surface it is often useful to derive normals for the resulting surface from the original normals. Specifically, given a 3×3 transformation matrix M , {\displaystyle \mathbf {M} ,} we can determine the matrix W {\displaystyle \mathbf {W} } that transforms a vector n {\displaystyle \mathbf {n} } perpendicular to

3395-945: Is a given scalar function . If F {\displaystyle F} is continuously differentiable then the hypersurface is a differentiable manifold in the neighbourhood of the points where the gradient is not zero. At these points a normal vector is given by the gradient: n = ∇ F ( x 1 , x 2 , … , x n ) = ( ∂ F ∂ x 1 , ∂ F ∂ x 2 , … , ∂ F ∂ x n ) . {\displaystyle \mathbb {n} =\nabla F\left(x_{1},x_{2},\ldots ,x_{n}\right)=\left({\tfrac {\partial F}{\partial x_{1}}},{\tfrac {\partial F}{\partial x_{2}}},\ldots ,{\tfrac {\partial F}{\partial x_{n}}}\right)\,.} The normal line

3492-516: Is a normal. For a plane whose equation is given in parametric form r ( s , t ) = r 0 + s p + t q , {\displaystyle \mathbf {r} (s,t)=\mathbf {r} _{0}+s\mathbf {p} +t\mathbf {q} ,} where r 0 {\displaystyle \mathbf {r} _{0}} is a point on the plane and p , q {\displaystyle \mathbf {p} ,\mathbf {q} } are non-parallel vectors pointing along

3589-399: Is a point on the hyperplane and p i {\displaystyle \mathbf {p} _{i}} for i = 1 , … , n − 1 {\displaystyle i=1,\ldots ,n-1} are linearly independent vectors pointing along the hyperplane, a normal to the hyperplane is any vector n {\displaystyle \mathbf {n} } in

3686-456: Is a vector perpendicular to the tangent plane of the surface at P . The word normal is also used as an adjective: a line normal to a plane , the normal component of a force , the normal vector , etc. The concept of normality generalizes to orthogonality ( right angles ). The concept has been generalized to differentiable manifolds of arbitrary dimension embedded in a Euclidean space . The normal vector space or normal space of

3783-396: Is by definition a surface normal. Alternatively, if the hyperplane is defined as the solution set of a single linear equation a 1 x 1 + ⋯ + a n x n = c , {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=c,} then the vector n = ( a 1 , … ,

3880-759: 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 )}

3977-451: 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

Rolling - Misplaced Pages Continue

4074-789: Is perpendicular to }}M\mathbb {t} \quad \,&{\text{ if and only if }}\quad 0=(W\mathbb {n} )\cdot (M\mathbb {t} )\\&{\text{ if and only if }}\quad 0=(W\mathbb {n} )^{\mathrm {T} }(M\mathbb {t} )\\&{\text{ if and only if }}\quad 0=\left(\mathbb {n} ^{\mathrm {T} }W^{\mathrm {T} }\right)(M\mathbb {t} )\\&{\text{ if and only if }}\quad 0=\mathbb {n} ^{\mathrm {T} }\left(W^{\mathrm {T} }M\right)\mathbb {t} \\\end{alignedat}}} Choosing W {\displaystyle \mathbf {W} } such that W T M = I , {\displaystyle W^{\mathrm {T} }M=I,} or W = ( M − 1 ) T , {\displaystyle W=(M^{-1})^{\mathrm {T} },} will satisfy

4171-640: 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

4268-488: 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,

4365-446: 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

4462-431: 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

4559-1037: 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

4656-437: 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

4753-413: 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

4850-831: Is the one-dimensional subspace with basis { n } . {\displaystyle \{\mathbf {n} \}.} A differential variety defined by implicit equations in the n {\displaystyle n} -dimensional space R n {\displaystyle \mathbb {R} ^{n}} is the set of the common zeros of a finite set of differentiable functions in n {\displaystyle n} variables f 1 ( x 1 , … , x n ) , … , f k ( x 1 , … , x n ) . {\displaystyle f_{1}\left(x_{1},\ldots ,x_{n}\right),\ldots ,f_{k}\left(x_{1},\ldots ,x_{n}\right).} The Jacobian matrix of

4947-411: 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

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5044-529: 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,

5141-418: 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,

5238-419: 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

5335-658: 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

5432-575: 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

5529-483: 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,

5626-532: 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

5723-446: The null space of the matrix P = [ p 1 ⋯ p n − 1 ] , {\displaystyle P={\begin{bmatrix}\mathbf {p} _{1}&\cdots &\mathbf {p} _{n-1}\end{bmatrix}},} meaning P n = 0 . {\displaystyle P\mathbf {n} =\mathbf {0} .} That is, any vector orthogonal to all in-plane vectors

5820-539: 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

5917-439: The above equation, giving a W n {\displaystyle W\mathbb {n} } perpendicular to M t , {\displaystyle M\mathbb {t} ,} or an n ′ {\displaystyle \mathbf {n} ^{\prime }} perpendicular to t ′ , {\displaystyle \mathbf {t} ^{\prime },} as required. Therefore, one should use

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6014-1558: 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

6111-595: 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

6208-419: The curve at the point. A normal vector of length one is called a unit normal vector . A curvature vector is a normal vector whose length is the curvature of the object. Multiplying a normal vector by −1 results in the opposite vector , which may be used for indicating sides (e.g., interior or exterior). In three-dimensional space , a surface normal , or simply normal , to a surface at point P

6305-421: 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

6402-404: 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,

6499-836: 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

6596-412: 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

6693-1857: The graph of a function z = f ( x , y ) , {\displaystyle z=f(x,y),} an upward-pointing normal can be found either from the parametrization r ( x , y ) = ( x , y , f ( x , y ) ) , {\displaystyle \mathbf {r} (x,y)=(x,y,f(x,y)),} giving n = ∂ r ∂ x × ∂ r ∂ y = ( 1 , 0 , ∂ f ∂ x ) × ( 0 , 1 , ∂ f ∂ y ) = ( − ∂ f ∂ x , − ∂ f ∂ y , 1 ) ; {\displaystyle \mathbf {n} ={\frac {\partial \mathbf {r} }{\partial x}}\times {\frac {\partial \mathbf {r} }{\partial y}}=\left(1,0,{\tfrac {\partial f}{\partial x}}\right)\times \left(0,1,{\tfrac {\partial f}{\partial y}}\right)=\left(-{\tfrac {\partial f}{\partial x}},-{\tfrac {\partial f}{\partial y}},1\right);} or more simply from its implicit form F ( x , y , z ) = z − f ( x , y ) = 0 , {\displaystyle F(x,y,z)=z-f(x,y)=0,} giving n = ∇ F ( x , y , z ) = ( − ∂ f ∂ x , − ∂ f ∂ y , 1 ) . {\displaystyle \mathbf {n} =\nabla F(x,y,z)=\left(-{\tfrac {\partial f}{\partial x}},-{\tfrac {\partial f}{\partial y}},1\right).} Since

6790-1038: The inverse transpose of the linear transformation when transforming surface normals. The inverse transpose is equal to the original matrix if the matrix is orthonormal, that is, purely rotational with no scaling or shearing. For an ( n − 1 ) {\displaystyle (n-1)} -dimensional hyperplane in n {\displaystyle n} -dimensional space R n {\displaystyle \mathbb {R} ^{n}} given by its parametric representation r ( t 1 , … , t n − 1 ) = p 0 + t 1 p 1 + ⋯ + t n − 1 p n − 1 , {\displaystyle \mathbf {r} \left(t_{1},\ldots ,t_{n-1}\right)=\mathbf {p} _{0}+t_{1}\mathbf {p} _{1}+\cdots +t_{n-1}\mathbf {p} _{n-1},} where p 0 {\displaystyle \mathbf {p} _{0}}

6887-408: 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

6984-464: 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

7081-542: 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

7178-458: The normal vector space at a point is the vector space generated by the normal vectors of the hypersurfaces at the point. The normal (affine) space at a point P {\displaystyle P} of the variety is the affine subspace passing through P {\displaystyle P} and generated by the normal vector space at P . {\displaystyle P.} These definitions may be extended verbatim to

7275-415: 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

7372-468: 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

7469-478: 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

7566-417: 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

7663-412: The obvious ambiguities of Einstein’s treatment, the notion of frame of reference has reappeared as a structure distinct from a coordinate system. Normal (geometry) In geometry , a normal is an object (e.g. a line , ray , or vector ) that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the line perpendicular to the tangent line to

7760-451: 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

7857-403: 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

7954-531: The plane, a normal to the plane is a vector normal to both p {\displaystyle \mathbf {p} } and q , {\displaystyle \mathbf {q} ,} which can be found as the cross product n = p × q . {\displaystyle \mathbf {n} =\mathbf {p} \times \mathbf {q} .} If a (possibly non-flat) surface S {\displaystyle S} in 3D space R 3 {\displaystyle \mathbb {R} ^{3}}

8051-401: 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

8148-396: The points where the variety is not a manifold. Let V be the variety defined in the 3-dimensional space by the equations x y = 0 , z = 0. {\displaystyle x\,y=0,\quad z=0.} This variety is the union of the x {\displaystyle x} -axis and the y {\displaystyle y} -axis. At a point (

8245-599: 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

8342-626: 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

8439-430: 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

8536-440: The rows of the Jacobian matrix are ( 0 , 0 , 1 ) {\displaystyle (0,0,1)} and ( 0 , 0 , 0 ) . {\displaystyle (0,0,0).} Thus the normal vector space and the normal affine space have dimension 1 and the normal affine space is the z {\displaystyle z} -axis. The normal ray is the outward-pointing ray perpendicular to

8633-467: 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

8730-484: 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

8827-426: The set of points ( x 1 , x 2 , … , x n ) {\displaystyle (x_{1},x_{2},\ldots ,x_{n})} satisfying an equation F ( x 1 , x 2 , … , x n ) = 0 , {\displaystyle F(x_{1},x_{2},\ldots ,x_{n})=0,} where F {\displaystyle F}

8924-450: The surface is given by the gradient n = ∇ F ( x , y , z ) . {\displaystyle \mathbf {n} =\nabla F(x,y,z).} since the gradient at any point is perpendicular to the level set S . {\displaystyle S.} For a surface S {\displaystyle S} in R 3 {\displaystyle \mathbb {R} ^{3}} given as

9021-1175: The tangent plane t {\displaystyle \mathbf {t} } into a vector n ′ {\displaystyle \mathbf {n} ^{\prime }} perpendicular to the transformed tangent plane M t , {\displaystyle \mathbf {Mt} ,} by the following logic: Write n′ as W n . {\displaystyle \mathbf {Wn} .} We must find W . {\displaystyle \mathbf {W} .} W n  is perpendicular to  M t  if and only if  0 = ( W n ) ⋅ ( M t )  if and only if  0 = ( W n ) T ( M t )  if and only if  0 = ( n T W T ) ( M t )  if and only if  0 = n T ( W T M ) t {\displaystyle {\begin{alignedat}{5}W\mathbb {n} {\text{

9118-395: 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} }

9215-434: The variety is the k × n {\displaystyle k\times n} matrix whose i {\displaystyle i} -th row is the gradient of f i . {\displaystyle f_{i}.} By the implicit function theorem , the variety is a manifold in the neighborhood of a point where the Jacobian matrix has rank k . {\displaystyle k.} At such

9312-416: 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

9409-542: 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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