Finding Mean Angular Velocity of Rotating Body

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SUMMARY

The discussion focuses on calculating the mean angular velocity of a solid body undergoing angular deceleration proportional to the square root of its angular velocity. The initial angular velocity is denoted as \varpi0, and the mean angular velocity is derived to be <\varpi> = \varpi0/3. The relevant equations include \varpif = \varpii + \betat and \varpiavg = \varphi/t, which are essential for solving the problem through integration.

PREREQUISITES
  • Understanding of angular kinematics, specifically angular velocity and deceleration.
  • Familiarity with calculus, particularly integration techniques.
  • Knowledge of the relationship between angular displacement, velocity, and time.
  • Basic grasp of proportional relationships in physics.
NEXT STEPS
  • Study the principles of angular motion and deceleration in physics.
  • Learn integration techniques for solving differential equations.
  • Explore the concept of mean values in rotational dynamics.
  • Review examples of angular velocity calculations in physics problems.
USEFUL FOR

Students studying physics, particularly those focusing on rotational dynamics, as well as educators looking for examples of angular velocity calculations in homework or teaching materials.

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Homework Statement


A solid body rotates with deceleration about a stationary axis with an angular deceleration
[tex]\beta[/tex] [tex]\propto[/tex] [tex]\sqrt{\varpi}[/tex] , where [tex]\varpi[/tex] is the angular velocity. Find the mean angular velocity of the body averaged over the whole time of the rotation if at the initial moment of time the angular velocity was [tex]\varpi[/tex]0

Homework Equations


[tex]\varpi[/tex]f = [tex]\varpi[/tex]i + [tex]\beta[/tex]t
[tex]\varpi[/tex]avg = [tex]\varphi[/tex]/t


The Attempt at a Solution


The given answer is <[tex]\varpi[/tex]> = [tex]\varpi[/tex]0/3
 
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, but I don't understand how to get there. We know that \varpif = \varpii + \betat, so we can integrate the expression \int_0^t (\varpii + \betat) dt. But then what?Can someone explain to me how to solve this?
 

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