Lecture Notes on Rotational Dynamics of a Centrifuge

Introduction

• A centrifuge is a rotating device or spinning disc.
• Key information:
  • Radius: 25 cm
  • Initial rotation: 4000 rpm (revolutions per minute)
  • Direction: Clockwise
  • Time to stop: 30 seconds
  • Tasks: Calculate average angular acceleration, tangential acceleration, net acceleration, total acceleration.

Terminology Recap

• Angular Acceleration: Change in angular speed over time.
• Tangential Acceleration: Depends on radius and angular acceleration.
• Centripetal Acceleration: Directed towards the center of rotation.
• Units: Revolution per minute (rpm) to Radian per second.

Calculations

Angular Speed Conversion

• Convert 4000 rpm to radian per second using:
  • Formula: ( \text{RPM to Rad} = \frac{2\pi}{60} )
  • Result: 419 rad/s

Average Angular Acceleration

• Formula: ( \Delta \omega / \Delta t )
• Calculations:
  • ( \Delta \omega = 0 - 419 )
  • Time: 30 seconds
  • Result: (-13.97 \text{ rad/s}^2) (negative indicates slowing down)

Tangential Acceleration

• Formula: ( r \times \alpha )
• Radius: 25 cm
• Result: 3.49 m/s²
• Note: Tangential acceleration depends on position.

Right-Hand Rule for Rotation

• Direction of angular speed using right-hand curl rule.
• Angular speed: Thumb points downwards (clockwise rotation)
• Angular acceleration: Upward (opposite slowing down)

Angular Speed at 29 Seconds

• Formula: ( \omega = \omega_i + \alpha t )
• Initial ( \omega = 419 \text{ rad/s} )
• ( \alpha = -13.97 \text{ rad/s}^2 )
• Result: 13.87 rad/s

Tangential Speed

• Formula: ( r \times \omega )
• Result: 3.47 m/s

Centripetal Acceleration

• Formula: ( v_t^2 / r )
• Tangential velocity: 3.47 m/s
• Result: 48.1 m/s²

Net Acceleration

• Formula: ( \sqrt{a_c^2 + a_t^2} )
• Result: 48.2 m/s²

Direction of Total Acceleration

• Use tangent theta for direction:
• Formula: ( \tan(\theta) = a_c / a_t )
• Result: 85.9 degrees (opposite to motion direction)

Conclusion

• Differences between angular and tangential acceleration.
• Net acceleration is primarily centripetal.
• Encouragement to watch other videos for terminology.

Additional Information

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