Rolling
Rolling is a
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
Applications
Most
One of the most practical applications of rolling objects is the use of
Rolling objects are also frequently used as
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 , where is the
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
Derivation Let 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. Since the center of mass has an immediate velocity as if it was rotating around the point of contact, its velocity is . Due to symmetry, the object center of mass is a point in its axis. Let be inertia of rotational inertia associated with rolling is (same as the rotational inertia of pure rotation around the point of contact). Using the general formula for kinetic energy of rotation, we have:
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Forces and acceleration
Differentiating the relation between linear and angular velocity, , with respect to time gives a formula relating linear and angular acceleration . Applying
It follows that to accelerate the object, both a net force and a
Derivation Assume that the object experiences an external force which exerts no torque (it has 0 moment arm ), static friction at the point of contact () provides the torque and the other forces involved cancel. is tangential to the object and surface at the point of contact and opposite in direction to . Using the sign convention by which this force is positive, the net force is:
Because there is no slip, holds. Substituting and for the linear and rotational version of Newton's second law, then solving for :
Expanding in : The last equality is the first formula for ; using it together with Newton's second law, then reducing, the formula for is obtained: The radius of gyration can be incorporated in the first formula for as follows: Substituting the latest equality above in the first formula for the second formula for it: |
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 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 . 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
In the specific case of an object rolling in an
Derivation Assuming that the object is placed so as to roll downward in the direction of the inclined plane (and not partially sideways), the weight can be decomposed in a component in the direction of rolling and a component perpendicular to the inclined plane. Only the first force component makes the object roll, the second is balanced by the contact force, but it does not form an action‐reaction pair with it (just as an object in rest on a table). Therefore, for this analysis, only the first component is considered, thus: In the last equality the denominator is the same as in the formula for force, but the factor disappears because its instance in the force of gravity cancels with its instance due to Newton's third law. |
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
References
- ISBN 9781118230718. Retrieved 13 January 2024.
- . Retrieved 28 December 2022.
- . Retrieved 28 December 2022.
See also
- Rolling resistance
- Frictional contact mechanics: Rolling contact
- Terrestrial locomotion in animals: Rolling
- Plantigrade
- Leg mechanism
- Tumbling (gymnastics)
- Roulette (curve)
- Trochoid
- Cycloid
- Gear
- Rack and pinion