Rolling on a surface which is moving with constant velocity
Rolling on a Surface Which is Moving with Constant Velocity
When we talk about rolling on a moving surface, we are considering a scenario where an object, typically a sphere or a cylinder, is rolling without slipping on a surface that itself is moving with a constant velocity. This situation is a bit more complex than rolling on a stationary surface because both the rolling object and the surface have their own velocities and the relative motion between them must be considered.
Key Concepts
Rolling Without Slipping
Rolling without slipping occurs when the point of contact between the rolling object and the surface is momentarily at rest with respect to the surface. This means that the linear velocity of the rolling object at the point of contact is equal to the velocity of the surface at that point.
For an object rolling without slipping, the relationship between its angular velocity ($\omega$) and its linear velocity ($v$) is given by:
$$ v = \omega r $$
where $r$ is the radius of the rolling object.
Moving Surface
When the surface is moving with a constant velocity ($v_s$), the condition for rolling without slipping is modified. The linear velocity of the rolling object at the point of contact must now match the velocity of the moving surface.
Relative Motion
The relative velocity ($v_{rel}$) of the rolling object with respect to the surface is the difference between the velocity of the object's center of mass ($v_{cm}$) and the velocity of the surface ($v_s$):
$$ v_{rel} = v_{cm} - v_s $$
Formulas
For an object rolling on a moving surface:
- The condition for rolling without slipping is:
$$ v_{cm} - v_s = \omega r $$
- The kinetic energy of the rolling object is the sum of its translational and rotational kinetic energies:
$$ KE = \frac{1}{2} m v_{cm}^2 + \frac{1}{2} I \omega^2 $$
where $m$ is the mass of the object and $I$ is its moment of inertia.
Differences and Important Points
| Aspect | Rolling on Stationary Surface | Rolling on Moving Surface with Velocity $v_s$ |
|---|---|---|
| Condition for No Slipping | $v_{cm} = \omega r$ | $v_{cm} - v_s = \omega r$ |
| Relative Velocity | $v_{rel} = 0$ | $v_{rel} = v_{cm} - v_s$ |
| Kinetic Energy | $\frac{1}{2} m v_{cm}^2 + \frac{1}{2} I \omega^2$ | $\frac{1}{2} m v_{cm}^2 + \frac{1}{2} I \omega^2$ |
| Dynamics | $F = ma$ | $F = ma$ (where $F$ is the net force, $m$ is the mass, and $a$ is the acceleration) |
| Frictional Force | Static friction acts to prevent slipping | Static friction acts to match the velocity of the rolling object with the moving surface |
Examples
Example 1: Rolling on a Conveyor Belt
Consider a cylinder rolling without slipping on a conveyor belt that is moving to the right with a constant velocity $v_s$. If the cylinder has a radius $r$ and its center of mass has a velocity $v_{cm}$ to the right, then the condition for rolling without slipping is:
$$ v_{cm} - v_s = \omega r $$
If the cylinder is initially at rest and the conveyor belt starts moving, static friction will act on the cylinder to accelerate it until it matches the speed of the conveyor belt, at which point rolling without slipping occurs.
Example 2: Bowling Ball on a Moving Walkway
A bowling ball is rolled on a moving walkway at an airport. If the walkway is moving with a constant velocity $v_s$ and the ball is rolled in the same direction with a center of mass velocity $v_{cm}$, the relative velocity of the ball with respect to the walkway is $v_{cm} - v_s$. The ball will continue to roll without slipping as long as this relative velocity matches the condition $v_{cm} - v_s = \omega r$.
In both examples, the key to understanding the motion is to consider the relative velocities and the condition for rolling without slipping. The dynamics of the rolling object are influenced by the moving surface, and the static frictional force plays a crucial role in establishing the rolling motion.