Rolling

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.

In practice, due to small deformations near the contact area, some sliding and energy dissipation occurs.

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.

Two well known non-axially-symmetrical rollers are the Reuleaux triangle and the Meissner bodies.

The construction of a specific surface allows even a perfect square wheel to roll with its centroid at constant height above a reference plane.

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.

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).

This is the basis for which almost all motors (such as those found in ceiling fans, cars, drills, etc.)

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.

One of the most basic ways is by placing a (usually flat) object on a series of lined-up rollers, or wheels.

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.

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.

[1] 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.

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

be the distance between the center of mass and the point of contact; when the surface is flat, this is the radius of the object around its widest cross section.

Due to symmetry, the object center of mass is a point in its axis.

, with respect to time gives a formula relating linear and angular acceleration

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.

for the linear and rotational version of Newton's second law, then solving for

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

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:

is specific to the object shape and mass distribution, it does not depend on scale or density.

When an axisymmetric deformable body contacts 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 a Lagrangian description of deformation.

[2][3] This approach greatly simplifies analysis by eliminating time-dependence, resulting in displacement, velocity, stress and strain fields that vary only spatially.