Finding Mean Angular Velocity of Rotating Body

In summary, the problem involves finding the mean angular velocity of a rotating solid body with deceleration about a stationary axis. The angular deceleration is proportional to the square root of the angular velocity, and the initial angular velocity is given as \varpi0. The solution is <\varpi> = \varpi0/3, but the process to get there involves integrating the expression \int_0^t (\varpii + \betat) dt. Further clarification is needed.
  • #1
paragchitnis
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0

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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  • #2
, 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?
 

1. What is mean angular velocity?

The mean angular velocity of a rotating body is the average rate at which the body is rotating over a given period of time. It is typically measured in radians per second or degrees per second.

2. How do you calculate mean angular velocity?

To calculate mean angular velocity, you divide the total change in angular displacement by the total change in time. This can be represented by the formula:
mean angular velocity = (Δθ/Δt), where Δθ is the change in angular displacement and Δt is the change in time.

3. What is the difference between mean angular velocity and instantaneous angular velocity?

Mean angular velocity is the average rate of rotation over a period of time, while instantaneous angular velocity is the rate of rotation at a specific moment in time. In other words, mean angular velocity gives an overall picture of the rotation, while instantaneous angular velocity gives a snapshot of the rotation at a particular point.

4. How does the mass and shape of a rotating body affect its mean angular velocity?

The mass and shape of a rotating body do not directly affect its mean angular velocity. However, they can indirectly impact it by affecting the body's moment of inertia, which is a measure of an object's resistance to change in rotation. Bodies with larger moments of inertia will have a lower mean angular velocity compared to bodies with smaller moments of inertia.

5. Can mean angular velocity be negative?

Yes, mean angular velocity can be negative. This indicates that the body is rotating in the opposite direction of the chosen reference direction. For example, a negative mean angular velocity would mean that the body is rotating clockwise instead of counterclockwise.

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