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.
No sliding takes place if and only if the instantaneous velocity of the rolling object in the point(s) in which it contacts the surface is the same as that of the surface; this is referred to as pure rolling. In particular, for a reference plane in which the rolling surface is at rest, the instantaneous velocity of the point of contact 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 most axially symmetrical 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 linear motion. Rolling objects are not necessarily axially-symmetrical. Two well known non-axially-symmetrical rollers are the Reuleaux triangle and the Meissner bodies. Objects with corners, such as dice, roll by successive rotations about the edge or corner which is in contact with the surface.
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. 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.
Physics of simple rolling
The simplest case of rolling is that of an axially symmetrical object rolling without slip along a flat surface with its axis parallel to the surface (or equivalently: perpendicular to the surface normal).
The velocity of any point in the rolling object is given by , where is the displacement between the particle and the rolling object's contact point (or line) with the surface, and 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.
Forces and acceleration
Differentiating the relation between linear and angular velocity, , with respect to time gives a formula relating linear and angular acceleration . Applying Newton's second law:
It follows that to accelerate 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 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:
has dimension of mass, and it is the mass that would have a rotational inertia at distance from an axis of rotation. Therefore, the term may be thought of the mass that has the equivalent linear inertia to the rolling object rotational inertia (among 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 . Similarly, the rotational inertia attenuates the effect of the external force by the dimensionless multiplicative factor to give a smaller net force.
In the specific case of an object rolling in an inclined plane which experiences only static friction, normal force and its own weight, the acceleration (in the direction of rolling down the slope) is:
Halliday, David; Resnick, Robert (2013), Fundamentals of Physics, Chapters 10, 11: Wiley