Rolling: Definition, Constraint, Mass, and More

[Greg Bernhardt]

Definition/Summary

"Rolling" means moving along a surface without sliding.

The (instantaneous) point of contact is stationary relative to the surface. In other words: it is the instantaneous centre of rotation (if that surface is stationary).

Friction at the point of contact of a rolling body is static friction.

A body rolling on a surface obeys the rolling constraint $v_{rel}\ =\ r\omega$ where $r$ is the instantaneous radius of curvature, $v_{rel}$ is the speed (relative to the surface) of the instantaneous centre of curvature, and $\omega$ is the instantaneous angular speed.

A rolling body with constant rolling radius has a "rolling mass" of $I/r^2$, equal to the moment of inertia divided by the square of the radius. In many circumstances, the rolling can be ignored provided that the ordinary (inertial) mass is increased by this "rolling mass".

Equations



Extended explanation

Rolling constraint:

Since there is no sliding, the point of contact must move the same distance (arc-length) along the body as along the surface. Together with the first and second derivatives, this gives the rolling constraints:

$ds_{rel} = r\,d\theta\ \ \ \ v_{rel} = r\omega\ \ \ \ a_{rel} = r\alpha$​

where $s\ v\ a,\ \theta\ \omega\ \alpha,\text{ and }r$ are distance speed and acceleration, angular distance speed and acceleration, and (instantaneous) radius of curvature.

Usually, the surface is not moving, and the subscript "rel" may be omittted.

The vector form of the rolling constraint is $\boldsymbol{v}_{rel} = \boldsymbol{\omega}\times\boldsymbol{r}\ \ \ \ \boldsymbol{a}_{rel} = \boldsymbol{\alpha}\times\boldsymbol{r}$

Rolling mass:

If a net force T parallel to a stationary surface acts at the centre of a rolling body of constant curvature, the equation for angular acceleration about the (stationary) point of contact, and the rolling constraint, gives:

$Tr = I_{p.o.c}\alpha = (mr^2 + I_{c.o.m})\alpha\ \ \ a_{rel} = a = r\alpha$​

and therefore:

$T = (m + I_{c.o.m}/r^2)a$​

exactly as if the (inertial) mass was increased by the "rolling mass" $I_{c.o.m}/r^2$, and there was no rolling or friction.

If a force is at a height h above the surface, or more generally if its line of action misses the point of contact by a distance h, then it is "geared" by a factor h/r: $T(h/r) = (m + I_{c.o.m}/r^2)a$. For example, if the surface is at a angle $\theta$ to the horizontal, then the weight mg misses the point of contact by $h = r\sin\theta$, and the effective acceleration due to gravity is only $g\sin\theta$.

We use the point of contact because the moment (torque) of the friction about that point is zero.

Note that we have not needed the linear "F = ma" equation.

Moving surface:

If the surface is moving with acceleration $a_s$, then the point of contact is not stationary, and so is not the centre of rotation, and we must use the more general formula for angular momentum:

$Tr = (I_{c.o.m})\alpha+mra\ \ \ a_{rel} = a - a_s = r\alpha$​

and therefore:

$T - ma_s = (m + I_{c.o.m}/r^2)a_{rel}$​

This is the standard equation for motion relative to a (non-inertial) frame fixed in the surface (with the standard "fictitious" inertial force $-ma_s$), and with the (inertial) mass again increased by the "rolling mass" $I_{c.o.m}/r^2$.

In the stationary (inertial) frame, this becomes:

$T + (I_{c.o.m}/r^2)a_s = (m + I_{c.o.m}/r^2)a$​

In particular, a body rolling freely (ie with zero T) on a moving surface has acceleration $I_{c.o.m}/(mr^2 + I_{c.o.m}) = I_{c.o.m}/I_{p.o.c}$ times the acceleration of the surface.

Massive pulley:

Similarly, a frictionless pulley over which a rope runs without slipping has the same effect as a body attached to the rope (with no other forces on it), with a mass equal to the rolling mass of the pulley.

In most exam questions, pulleys are light, meaning not massive, so that they have zero rolling mass and do not affect the system.

Friction:

Friction at the point of contact of a rolling body is static friction.

There is also a rolling resistance, or "rolling friction": an additional small force, caused by deformation of the body (in exam questions it can usually be ignored).

Friction at the point of contact of a sliding body is of course dynamic friction.

Sliding then rolling:

A stationary ball struck sharply sufficiently above the midline will start to move with "topspin": its ratio of angular speed to linear speed will be too high for rolling ($\omega/v > 1/r$): at the point of contact, it will be moving backward relative to the surface, so the (dynamic) friction force on it will be forward, therefore increasing its linear speed but decreasing its angular speed. This will continue until $\omega/v = 1/r$, when the ball will start rolling.

Similarly, a ball struck below the midline will start to move with "backspin" (with ($\omega/v < 1/r$)): its linear speed will decrease but it angular speed will increase, until the ball starts rolling.

Spools and pulleys:

A spool is a body with two rolling surfaces with different radii. Usually one surface rolls along a rigid body, and the other rolls along a string.

A fixed pulley can be treated as a spool whose "solid" radius is zero (because it not only rolls along the string, but can also be regarded as rolling along the axle, with $v_{rel}\ =\ r\ =\ 0$).

"Rolling" in ships and aircraft:

"rolling" has a different meaning in ships and aircraft: it means rotation about an axis parallel to the longitudinal axis of the vessel (usually the direction of motion)

so, in that context, a coin rolling back and forward inside a curved bowl is pitching, but not rolling (and if it was spinning, that would be yawing)

(in dancing, rocking 'n' rolling is mostly yawing)

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

 

[APhysicist]

Rolling is a type of motion that involves an object moving along a surface without sliding. The point of contact between the object and the surface is stationary relative to the surface, and the friction at this point is static. In order for a body to roll, it must obey the rolling constraint, which states that the speed (relative to the surface) of the instantaneous centre of curvature must be equal to the product of the instantaneous radius of curvature and the instantaneous angular speed. Additionally, the mass of a body rolling with constant rolling radius is increased by its "rolling mass," which is equal to the moment of inertia divided by the square of the radius. Rolling also has different meanings in ships and aircraft, as well as in dancing.