Rolling

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 \mathbf{v} = \boldsymbol{\omega}\times\mathbf{r}, where \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_\text{rolling} = K_\text{translation} + K_\text{rotation} Forces and acceleration Differentiating the relation between linear and angular velocity, v_\text{c.o.m.} = r\omega, with respect to time gives a formula relating linear and angular acceleration a = r\alpha. Applying Newton's second law: a = \frac{F_\text{net}}{m} = r\alpha = \frac{r\tau}{I}. 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: \begin{align} 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{align} \tfrac{I}{r^2} 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{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 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{I}{r^2}. 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 multiplicative factor 1/\left(1+\tfrac{I}{mr^2}\right) where \tfrac{I}{mr^2} represents the ratio of the aforesaid virtual mass to the object actual mass and it is equal to \left({r_\text{g}}/{r}\right)^2 where 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 is due to the fact rotational inertia of a point mass varies proportionally to the square of its distance to the axis. , animated GIF version. 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 = \frac{g\sin(\theta)}{1+\left({r_\text{g}}/{r}\right)^2} {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 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. == Deformable bodies ==

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. 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. == See also ==