Lecture on Rotational Dynamics: Rotating Fan

Key Concepts

• Angular Acceleration: A constant rate of speeding up, analogous to linear acceleration.
• Angular Speed (Omega): The rate at which the fan is rotating, measured in radians per second.
• Angular Displacement (Theta): The total angle in radians through which a point or line has been rotated in a specified sense about a specified axis.
• Revolutions: Complete turns made by the fan.

Problem Statement

• Initial Conditions:
  • Initial angular speed ( \omega_0 = 4 ) rad/s at ( t = 0 ) seconds.
  • Angular acceleration ( \alpha = 5 ) rad/s².
  • Time ( t = 10 ) seconds.

Calculations

Angular Speed at ( t = 10 ) seconds

• Formula: [ \omega = \omega_0 + \alpha \times t ]
• Substituting Values:
  • ( \omega_0 = 4 ) rad/s
  • ( \alpha = 5 ) rad/s²
  • ( t = 10 ) s
  • ( \omega = 4 + (5 \times 10) = 54 ) rad/s

Angular Displacement ( \Theta )

• Formula: [ \theta = \omega_0 \times t + \frac{1}{2} \alpha t^2 ]
• Substituting Values:
  • ( \omega_0 = 4 ) rad/s
  • ( \alpha = 5 ) rad/s²
  • ( t = 10 ) s
  • ( \theta = 4 \times 10 + \frac{1}{2} \times 5 \times 10^2 = 40 + 250 = 290 ) rad

Number of Revolutions

• Conversion:
  • 1 revolution = ( 2\pi ) radians
  • Total revolutions ( = \frac{290}{2\pi} \approx 46.2 )

Conclusion

• At ( t = 10 ) seconds, the angular speed is calculated to be 54 rad/s.
• The angular displacement is 290 radians.
• The fan completes approximately 46.2 revolutions.

Additional Information

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