Rolling on a stationary horizontal surface
Rolling on a Stationary Horizontal Surface
Rolling is a type of motion that involves both translational and rotational movements. When an object rolls on a stationary horizontal surface, it moves forward without slipping, meaning there is no relative motion between the point of contact on the rolling object and the surface.
Key Concepts
1. Rolling Without Slipping
When an object rolls without slipping, the velocity of the point of contact with the surface is zero relative to the surface. This condition can be expressed as:
$$ v = \omega r $$
where:
- $v$ is the translational velocity of the center of mass of the rolling object,
- $\omega$ is the angular velocity,
- $r$ is the radius of the rolling object.
2. Kinetic Energy in Rolling Motion
The total kinetic energy (KE) of a rolling object is the sum of its translational and rotational kinetic energies:
$$ KE = KE_{translational} + KE_{rotational} $$ $$ KE = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 $$
where:
- $m$ is the mass of the object,
- $I$ is the moment of inertia of the object about the axis of rotation.
3. Moment of Inertia
The moment of inertia depends on the shape of the object and the distribution of mass around the axis of rotation. For common shapes, the moment of inertia is given by standard formulas, such as:
- For a solid sphere: $I = \frac{2}{5}mr^2$
- For a solid cylinder or disk: $I = \frac{1}{2}mr^2$
- For a hollow cylinder: $I = mr^2$
4. Acceleration and Forces
When an object rolls under the influence of an external force, such as gravity or friction, it may experience linear acceleration ($a$) and angular acceleration ($\alpha$). These are related by:
$$ a = \alpha r $$
The force of friction is often the force that enables rolling without slipping. It acts at the point of contact and provides the torque necessary for rotation.
Differences and Important Points
| Aspect | Translational Motion | Rotational Motion | Rolling Motion |
|---|---|---|---|
| Type of Movement | Linear | Circular | Combination of both |
| Energy | Kinetic ($\frac{1}{2}mv^2$) | Rotational ($\frac{1}{2}I\omega^2$) | Sum of both |
| Velocity of Contact Point | Equal to $v$ | Zero at the axis, increases with distance from axis | Zero relative to the surface |
| Acceleration | $a$ (linear) | $\alpha$ (angular) | Related by $a = \alpha r$ |
| Force of Friction | Opposes motion | Provides torque for rotation | Prevents slipping |
Formulas
- Velocity of the center of mass: $v = \omega r$
- Acceleration of the center of mass: $a = \alpha r$
- Total kinetic energy: $KE = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$
Examples
Example 1: Rolling Sphere
A solid sphere of radius $r$ rolls without slipping on a horizontal surface. If it has a translational velocity $v$, its angular velocity $\omega$ is given by:
$$ \omega = \frac{v}{r} $$
The total kinetic energy is:
$$ KE = \frac{1}{2}mv^2 + \frac{1}{2}\left(\frac{2}{5}mr^2\right)\left(\frac{v}{r}\right)^2 $$ $$ KE = \frac{1}{2}mv^2 + \frac{1}{2}\frac{2}{5}mv^2 $$ $$ KE = \frac{1}{2}mv^2 + \frac{1}{5}mv^2 $$ $$ KE = \frac{7}{10}mv^2 $$
Example 2: Rolling Cylinder
A solid cylinder of radius $r$ and mass $m$ rolls without slipping down an incline. The force of friction provides the necessary torque for rolling without slipping. The moment of inertia for a cylinder is $I = \frac{1}{2}mr^2$, and the total kinetic energy as it rolls is:
$$ KE = \frac{1}{2}mv^2 + \frac{1}{2}\left(\frac{1}{2}mr^2\right)\left(\frac{v}{r}\right)^2 $$ $$ KE = \frac{1}{2}mv^2 + \frac{1}{4}mv^2 $$ $$ KE = \frac{3}{4}mv^2 $$
Understanding the principles of rolling motion is crucial for solving problems related to rotational dynamics and is often tested in physics exams.