Pure rotational motion
Pure Rotational Motion
Pure rotational motion refers to the movement of a rigid body that spins around a fixed axis without undergoing any translation, meaning the body does not move from one place to another. Every point on the body follows a circular path, and all the points on the body have the same angular velocity and angular acceleration.
Key Concepts
Rigid Body
A rigid body is an idealization where the distance between any two points on the body remains constant despite the application of external forces or torques.
Rotation Axis
The rotation axis is the straight line that every point on the rotating body moves around. This axis can be internal or external to the body.
Angular Displacement
Angular displacement ($\theta$) is the angle through which a point or line has been rotated in a specified sense about a specified axis.
Angular Velocity
Angular velocity ($\omega$) is the rate of change of angular displacement and is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating.
Angular Acceleration
Angular acceleration ($\alpha$) is the rate of change of angular velocity.
Formulas
- Angular Displacement: $\theta = \theta_f - \theta_i$
- Angular Velocity: $\omega = \frac{d\theta}{dt}$
- Angular Acceleration: $\alpha = \frac{d\omega}{dt}$
Differences and Important Points
| Aspect | Linear Motion | Pure Rotational Motion |
|---|---|---|
| Motion Type | Translation along a straight line. | Rotation around a fixed axis. |
| Displacement | Linear displacement (meters, m). | Angular displacement (radians, rad). |
| Velocity | Linear velocity (meters/second, m/s). | Angular velocity (radians/second, rad/s). |
| Acceleration | Linear acceleration (m/s²). | Angular acceleration (rad/s²). |
| Inertia | Mass (kg). | Moment of inertia (kg·m²). |
| Force/Torque | Force (Newtons, N). | Torque (Newton-meters, N·m). |
| Equations of Motion | $s = ut + \frac{1}{2}at^2$ | $\theta = \omega_i t + \frac{1}{2}\alpha t^2$ |
| Newton's Second Law | $F = ma$ | $\tau = I\alpha$ |
Examples
Example 1: Rotating Wheel
Consider a wheel rotating about its axis. The wheel undergoes pure rotational motion because all points on the wheel move in circles around the axis, and the wheel itself does not translate from one location to another.
Example 2: Earth's Rotation
The Earth rotates around its axis, which is an example of pure rotational motion. All points on the Earth's surface move in circular paths around the axis, which passes through the North and South Poles.
Example 3: Moment of Inertia Calculation
The moment of inertia (I) of a solid cylinder rotating about its central axis is given by:
$$ I = \frac{1}{2}MR^2 $$
where $M$ is the mass of the cylinder and $R$ is the radius.
Example 4: Angular Acceleration
A spinning figure skater pulls in her arms and spins faster. This is due to conservation of angular momentum, but it also illustrates that the skater's angular acceleration increases as her moment of inertia decreases.
Example 5: Torque and Angular Acceleration
If a torque $\tau$ is applied to a rigid body with a moment of inertia $I$, the angular acceleration $\alpha$ can be found using:
$$ \tau = I\alpha $$
This is analogous to Newton's second law for linear motion ($F = ma$), where force is replaced by torque and mass by moment of inertia.
Understanding pure rotational motion is crucial for solving problems in mechanics involving rotating bodies, such as wheels, gears, and celestial bodies. It is also essential for engineering applications like the design of rotating machinery and for understanding phenomena in the natural world, such as the rotation of planets and the dynamics of spinning objects.