Introduction to Rotational Motion and Angular Displacement

Agenda

1. What is Rotational Motion?
2. Types of Rotational Motion
3. Angular Displacement
4. Angular Velocity
5. Angular Acceleration
6. Torque
7. Moment of Inertia
8. Key Formulas
9. Relationship Between Linear and Angular Quantities
10. Visualizing Rotational Motion
11. Applications of Rotational Motion
12. Conclusion
13. Q&A

1. What is Rotational Motion?

• Definition: Movement of an object around a fixed axis.
• Examples: Earth's rotation on its axis, spinning a top.

2. Types of Rotational Motion

• Rigid Body Rotation: All points of the object move in circles with the same axis.
• Non-Rigid Body Rotation: Different parts of the object may have different axes and radii.

3. Angular Displacement

• Definition: Measure of how far an object has rotated.
• Units: Degrees or Radians.
• Formula: $\theta =\frac{s}{r}$
  • Where $\theta$ is the angular displacement, $s$ is the arc length, and $r$ is the radius.

4. Angular Velocity

• Definition: Rate of change of angular displacement.
• Units: Radians per second.
• Formula: $\omega =\frac{\theta }{t}$
  • Where $\omega$ is the angular velocity, $\theta$ is the angular displacement, and $t$ is the time.

5. Angular Acceleration

• Definition: Rate of change of angular velocity.
• Units: Radians per second squared.
• Formula: $\alpha =\frac{\mathrm{\Delta }\omega }{\mathrm{\Delta }t}$
  • Where $\alpha$ is the angular acceleration, $\mathrm{\Delta }\omega$ is the change in angular velocity, and $\mathrm{\Delta }t$ is the change in time.

6. Torque

• Definition: Rotational equivalent of force.
• Units: Newton-meters (Nm).
• Formula: $\tau =rF\sin (\theta )$
  • Where $\tau$ is the torque, $r$ is the radius, $F$ is the force, and $\theta$ is the angle between the force vector and the lever arm.

7. Moment of Inertia

• Definition: Resistance of an object to rotational acceleration.
• Units: Kilogram-meters squared (kg · m²).
• Formula: $I=\sum m_i{r}_i^2$
  • Where $I$ is the moment of inertia, $m_i$ is the mass of each particle, and $r_i$ is the distance of each particle from the axis of rotation.

8. Key Formulas

• Angular Displacement: $\theta =\frac{s}{r}$
• Angular Velocity: $\omega =\frac{\theta }{t}$
• Angular Acceleration: $\alpha =\frac{\mathrm{\Delta }\omega }{\mathrm{\Delta }t}$
• Torque: $\tau =rF\sin (\theta )$
• Moment of Inertia: $I=\sum m_i{r}_i^2$

9. Relationship Between Linear and Angular Quantities

• Linear Velocity: $v=r\omega$
  • Where $v$ is the linear velocity, $r$ is the radius, and $\omega$ is the angular velocity.
• Centripetal Acceleration: $a_c={\omega }^{2}r$
  • Where $a_c$ is the centripetal acceleration, $\omega$ is the angular velocity, and $r$ is the radius.

10. Visualizing Rotational Motion

• Use diagrams to show initial and final positions.
• Animations can help visualize continuous motion.
• Vector diagrams to show direction of angular velocity and torque.

11. Applications of Rotational Motion

• Mechanical Systems: Gears, turbines, engines.
• Astronomy: Planetary motion, orbits, celestial mechanics.
• Biology: Joint rotations in the human body.

12. Example Problems

• Calculate angular displacement for a car tire.
• Determine angular velocity of a spinning wheel.
• Find the torque required to lift a weight using a lever.

Conclusion

• Rotational motion is a fundamental concept in physics.
• Understanding angular displacement, velocity, acceleration, and torque is crucial for analyzing rotational systems.

Q&A

• Are there any questions about rotational motion and angular displacement?
• Feel free to reach out for further discussions.