Rolling is a type of motion that combines rotation (commonly, of an axially symmetric object) and translation 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 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 (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 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 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 a reference plane.
Applications 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 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, 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 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 is used to apply normal forces to a moving line of contact in various processes, for example in metalworking, printing, rubber manufacturing, painting.
Rigid bodies
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, 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} } 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 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 is entirely a function of an object mass and velocity, the above result may be used with the parallel axis theorem to obtain the kinetic energy associated with simple rolling
K rolling = K translation + K rotation {\displaystyle K_{\text{rolling}}=K_{\text{translation}}+K_{\text{rotation}}}
Forces and acceleration Differentiating 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:
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