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Rolling is a type of motion that combines
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
(commonly, of an
axially symmetric In geometry, circular symmetry is a type of continuous symmetry for a Plane (geometry), planar object that can be rotational symmetry, rotated by any arbitrary angle and map onto itself. Rotational circular symmetry is isomorphic with the circl ...
object) and
translation Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
of that object with respect to a surface (either one or the other moves), such that, if ideal conditions exist, the two are in contact with each other without sliding. 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 physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both mathema ...
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 the points of contact (e.g., 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 Rolling resistance, sometimes called rolling friction or rolling drag, is the force resisting the motion when a body (such as a ball, tire, or wheel) rolls on a surface. It is mainly caused by non-elastic effects; that is, not all the energy nee ...
is much lower than
sliding friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of ...
, and thus, rolling objects, typically require much less
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
to be moved than sliding ones. As 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 In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weight function, weighted relative position (vector), position of the distributed mass sums to zero. Thi ...
performs a
circular motion In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of ro ...
, rather than a
linear motion Linear motion, also called rectilinear motion, is one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion, with co ...
. Rolling objects are not necessarily axially-symmetrical. Two well known non-axially-symmetrical rollers are the Reuleaux triangle and the Meissner bodies. The
oloid An oloid is a three-dimensional curved geometric object that was discovered by Paul Schatz in 1929. It is the convex hull of a skeletal frame made by placing two linked congruent circles in perpendicular planes, so that the center of each circl ...
and the
sphericon In solid geometry, the sphericon is a solid that has a continuous developable surface with two congruent, semi-circular edges, and four vertices that define a square. It is a member of a special family of rollers that, while being rolled on ...
are members of a special family of developable rollers that
develop Develop or DEVELOP may refer to: * ''Develop'' (magazine), a trade publication for the video game industry * ''Develop'' (Apple magazine), a technical magazine formerly published by Apple Computer * Develop (chess), moving a piece from its origina ...
their entire surface when rolling down a flat plane. Objects with corners, such as
dice Dice (singular die or dice) are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating random values, commonly as part of tabletop games, including dice games, board games, role-playing g ...
, 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 A square wheel is a wheel that, instead of being circular, has the shape of a square. While literal square wheels exist, a more common use is as an idiom meaning feeling bad and naive (see reinventing the wheel). A square wheel can roll smoothly ...
to roll with its centroid at constant height above a reference plane.


Applications

Most
land vehicle A vehicle (from la, vehiculum) is a machine that transports people or cargo. Vehicles include wagons, bicycles, motor vehicles (motorcycles, cars, trucks, buses, mobility scooters for disabled people), railed vehicles (trains, trams), wate ...
s use wheels and therefore rolling for displacement.
Slip Slip or SLIP may refer to: Science and technology Biology * Slip (fish), also known as Black Sole * Slip (horticulture), a small cutting of a plant as a specimen or for grafting * Muscle slip, a branching of a muscle, in anatomy Computing and ...
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 is the use of
rolling-element bearing In mechanical engineering, a rolling-element bearing, also known as a rolling bearing, is a bearing which carries a load by placing rolling elements (such as balls or rollers) between two concentric, grooved rings called races. The relative m ...
s, such as
ball bearings A ball bearing is a type of rolling-element bearing that uses balls to maintain the separation between the bearing races. The purpose of a ball bearing is to reduce rotational friction and support radial and axial loads. It achieves this ...
, 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 Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of t ...
. This is the basis for which almost all
motors An engine or motor is a machine designed to convert one or more forms of energy into mechanical energy. Available energy sources include potential energy (e.g. energy of the Earth's gravitational field as exploited in hydroelectric power gen ...
(such as those found in ceiling fans, cars, drills, etc.) rely on to operate. Alternatively, 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 An axle or axletree is a central shaft for a rotating wheel or gear. On wheeled vehicles, the axle may be fixed to the wheels, rotating with them, or fixed to the vehicle, with the wheels rotating around the axle. In the former case, bearing ...
. 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 A tool is an object that can extend an individual's ability to modify features of the surrounding environment or help them accomplish a particular task. Although many animals use simple tools, only human beings, whose use of stone tools dates ba ...
for
transportation Transport (in British English), or transportation (in American English), is the intentional movement of humans, animals, and goods from one location to another. Modes of transport include air, land (rail and road), water, cable, pipeline, ...
. One of the most basic ways is by placing a (usually flat) object on a series of lined-up rollers, or
wheel A wheel is a circular component that is intended to rotate on an axle Bearing (mechanical), bearing. The wheel is one of the key components of the wheel and axle which is one of the Simple machine, six simple machines. Wheels, in conjunction wi ...
s. 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 A car or automobile is a motor vehicle with wheels. Most definitions of ''cars'' say that they run primarily on roads, seat one to eight people, have four wheels, and mainly transport people instead of goods. The year 1886 is regarded as t ...
,
trains In rail transport, a train (from Old French , from Latin , "to pull, to draw") is a series of connected vehicles that run along a railway track and transport people or freight. Trains are typically pulled or pushed by locomotives (often know ...
, and other human transportation vehicles.


Physics of simple rolling

The simplest case of rolling is that of rolling without slipping along a flat surface with its axis parallel to the surface (or equivalently: perpendicular to the surface
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
). The trajectory of any point is a
trochoid In geometry, a trochoid () is a roulette curve formed by a circle rolling along a line. It is the curve traced out by a point fixed to a circle (where the point may be on, inside, or outside the circle) as it rolls along a straight line. If the ...
; in particular, 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 In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve ...
. The velocity of any point in the rolling object is given by \mathbf=\boldsymbol\times\mathbf, where \mathbf is the
displacement Displacement may refer to: Physical sciences Mathematics and Physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
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 Rotation around a fixed axis is a special case of rotational motion. The fixed-axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rota ...
, 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 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).


Energy

Since
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
is entirely a function of an object mass and velocity, the above result may be used with the
parallel axis theorem The parallel axis theorem, also known as Huygens–Steiner theorem, or just as Steiner's theorem, named after Christiaan Huygens and Jakob Steiner, can be used to determine the moment of inertia or the second moment of area of a rigid body abo ...
to obtain the kinetic energy associated with simple rolling : K_\text=K_\text+K_\text


Forces and acceleration

Differentiating the relation between linear and angular ''velocity'', v_\text=r\omega, with respect to time gives a formula relating linear and angular ''acceleration'' a=r\alpha. Applying
Newton's second law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...
: :a=\frac=r\alpha=\frac. It follows that to accelerate the object, both a net force and a
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
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 Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of t ...
, for example, between the road and a wheel or between a bowling lane and a bowling ball. When static friction isn't enough, the friction becomes
dynamic friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding (motion), sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative la ...
and slipping happens. The tangential force is opposite in direction to the external force, and therefore partially cancels it. The resulting
net force Net Force may refer to: * Net force, the overall force acting on an object * ''NetForce'' (film), a 1999 American television film * Tom Clancy's Net Force, a novel series * Tom Clancy's Net Force Explorers Tom Clancy's Net Force Explorers or Ne ...
and acceleration are: :\begin F_\text &= \frac = \frac \\ a &= \frac \end \tfrac has dimension of mass, and it is the mass that would have a rotational inertia I at distance r from an axis of rotation. Therefore, the term \tfrac 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 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+\tfrac. Since the work done by the external force is split between overcoming the translational and rotational inertia, the external force results in a smaller net force by the
dimensionless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
multiplicative factor 1/\left(1+\tfrac\right) where \tfrac represents the ratio of the aforesaid virtual mass to the object actual mass and it is equal to \left(\tfrac\right)^2 where r_\text is the
radius of gyration ''Radius of gyration'' or gyradius of a body about the axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body's actual distribution of mass, if the total mass of the body were concentr ...
corresponding to the object rotational inertia in pure rotation (not the rotational inertia in pure rolling). The square power 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 An inclined plane, also known as a ramp, is a flat supporting surface tilted at an angle from the vertical direction, with one end higher than the other, used as an aid for raising or lowering a load. The inclined plane is one of the six clas ...
which experiences only static friction, normal force and its own weight, (
air drag In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding ...
is absent) the acceleration in the direction of rolling down the slope is: :a=\frac \tfrac is specific to the object shape and mass distribution, it does not depend on scale or density. However, it will vary if 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.


References

{{citation, last1=Halliday, first1=David, last2=Resnick, first2=Robert, title=Fundamentals of Physics, date=2014, publisher=Wiley, location=Chapters 9


See also

*
Rolling resistance Rolling resistance, sometimes called rolling friction or rolling drag, is the force resisting the motion when a body (such as a ball, tire, or wheel) rolls on a surface. It is mainly caused by non-elastic effects; that is, not all the energy nee ...
* Frictional contact mechanics: Rolling contact * Terrestrial locomotion in animals: Rolling *
Plantigrade 151px, Portion of a human skeleton, showing plantigrade habit In terrestrial animals, plantigrade locomotion means walking with the toes and metatarsals flat on the ground. It is one of three forms of locomotion adopted by terrestrial mammals. T ...
* Leg mechanism *
Tumbling (gymnastics) Tumbling, sometimes referred to as power tumbling, is a gymnastics discipline in which participants perform a series of acrobatic skills down a long Sprung floor, sprung track. Each series, known as a pass, comprises eight elements in which the ...
*
Roulette (curve) In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, epitrochoids, hypotrochoids, and involutes. Definition Informal definition Roughly speaking, a roulette i ...
*
Trochoid In geometry, a trochoid () is a roulette curve formed by a circle rolling along a line. It is the curve traced out by a point fixed to a circle (where the point may be on, inside, or outside the circle) as it rolls along a straight line. If the ...
*
Cycloid In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve ...
*
Gear A gear is a rotating circular machine part having cut teeth or, in the case of a cogwheel or gearwheel, inserted teeth (called ''cogs''), which mesh with another (compatible) toothed part to transmit (convert) torque and speed. The basic pr ...
*
Rack and pinion A rack and pinion is a type of linear actuator that comprises a circular gear (the '' pinion'') engaging a linear gear (the ''rack''). Together, they convert rotational motion into linear motion. Rotating the pinion causes the rack to be driven ...
Rotation Articles containing video clips