Rotational Motion Revision Notes BTER Polytechnic Notes

 

Rotational Motion

Rotational motion deals with the motion of objects that rotate around a fixed point or axis. Understanding the basic concepts of rotational motion is essential for analyzing systems like wheels, gears, and planetary motion in physics and engineering.

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3.1 Circular Motion

Circular motion is the motion of an object along the circumference of a circle. It occurs when an object moves around a central point or axis with a constant distance from the center (radius).

3.1.1 Definition of Angular Displacement

• Angular Displacement is the angle through which an object moves along a circular path in a given time. It measures the change in the angle with respect to a reference point and is a vector quantity.

• Formula:

  $$\theta = \frac{s}{r}$$

  Where:

  • $\theta$ is the angular displacement (in radians),
  • $s$ is the arc length (distance traveled along the circular path),
  • $r$ is the radius of the circle.

• Units: The standard unit of angular displacement is radians (rad).

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3.1.2 Angular Velocity, Angular Acceleration, Frequency, and Time Period

1. Angular Velocity ($\omega$):

   • Definition: Angular velocity is the rate at which an object changes its angular position with respect to time. It is a vector quantity.

   • Formula:

     $$\omega = \frac{\Delta \theta}{\Delta t}$$

     Where:

     • $\omega$ is angular velocity (in radians per second, rad/s),
     • $\Delta \theta$ is the change in angular displacement (in radians),
     • $\Delta t$ is the time interval.

   • Relationship with Linear Velocity:

     $$v = r \cdot \omega$$

     Where:

     • $v$ is the linear velocity (m/s),
     • $r$ is the radius (m),
     • $\omega$ is the angular velocity (rad/s).

2. Angular Acceleration ($\alpha$):

   • Definition: Angular acceleration is the rate of change of angular velocity with respect to time.

   • Formula:

     $$\alpha = \frac{\Delta \omega}{\Delta t}$$

     Where:

     • $\alpha$ is angular acceleration (rad/s²),
     • $\Delta \omega$ is the change in angular velocity (rad/s),
     • $\Delta t$ is the time interval.

   • Equation of Motion:

     $$\omega_f = \omega_i + \alpha t$$

     Where:

     • $\omega_f$ is the final angular velocity,
     • $\omega_i$ is the initial angular velocity,
     • $\alpha$ is angular acceleration,
     • $t$ is the time.

3. Frequency (f):

   • Definition: Frequency is the number of complete rotations or cycles per unit time.
   • Formula: $$f = \frac{1}{T}$$ Where:
     • $f$ is the frequency (in hertz, Hz),
     • $T$ is the time period (in seconds).

4. Time Period (T):

   • Definition: The time period is the time taken for one complete revolution or cycle of an object moving in circular motion.
   • Formula: $$T = \frac{2\pi}{\omega}$$ Where:
     • $T$ is the time period (in seconds),
     • $\omega$ is the angular velocity (rad/s).

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3.2 Centripetal and Centrifugal Forces with Live Examples

Centripetal Force:

• Definition: Centripetal force is the force that acts on an object moving in a circular path, directed towards the center of the circle or the axis of rotation. It keeps the object moving in a circular trajectory.

• Formula:

  $$F_{\text{centripetal}} = \frac{m v^2}{r} = m \omega^2 r$$

  Where:

  • $F_{\text{centripetal}}$ is the centripetal force (in newtons, N),
  • $m$ is the mass of the object (in kg),
  • $v$ is the linear velocity (in m/s),
  • $r$ is the radius of the circular path (in meters),
  • $\omega$ is the angular velocity (rad/s).

• Examples of Centripetal Force:

  1. Car Turning Around a Curve: When a car turns around a bend in the road, the friction between the tires and the road provides the centripetal force that keeps the car from skidding outward.
  2. Planetary Motion: The gravitational force between the Sun and the planets provides the centripetal force that keeps the planets in orbit around the Sun.

Centrifugal Force:

• Definition: Centrifugal force is the apparent force that seems to push an object away from the center of a circular path. It is not a real force but is experienced in a rotating reference frame, and it arises due to the inertia of the object.

• Formula:

  $$F_{\text{centrifugal}} = m \cdot \omega^2 \cdot r$$

  Where:

  • $F_{\text{centrifugal}}$ is the centrifugal force (in newtons, N),
  • $m$ is the mass of the object (in kg),
  • $\omega$ is the angular velocity (rad/s),
  • $r$ is the radius (in meters).

• Examples of Centrifugal Force:

  1. Spinning in a Washing Machine: In a washing machine during the spin cycle, clothes experience centrifugal force that causes them to move outward, away from the center of the drum.
  2. Riding a Merry-Go-Round: When riding a merry-go-round, you feel pushed outward. This is the centrifugal force, which is the reaction to the centripetal force that is acting towards the center.

Difference Between Centripetal and Centrifugal Forces:

• Centripetal Force: A real force, directed towards the center of the circular path.
• Centrifugal Force: An apparent force that acts outward, felt in a rotating reference frame, and is a result of inertia.

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Summary of Key Concepts:

1. Circular Motion: The motion of an object along the circumference of a circle.
2. Angular Displacement: The angle through which an object moves along a circular path.
3. Angular Velocity: The rate of change of angular displacement.
4. Angular Acceleration: The rate of change of angular velocity.
5. Frequency and Time Period: Frequency is the number of cycles per second, and the time period is the time taken for one complete cycle.
6. Centripetal Force: The force that keeps an object in circular motion, directed towards the center of the circle.
7. Centrifugal Force: The apparent force that pushes objects away from the center of rotation, experienced in a rotating reference frame.

These concepts are essential for understanding the dynamics of rotating objects and are applicable in real-world scenarios, including planetary motion, vehicle dynamics, and machinery with rotating parts.