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Image Added

Spinning Top
from Wikimedia Commons: Image by User:Lacen

Composition Setup
 Excerpt
The rapidly spinning Symmetric Top exhibiting Precession under the force of gravity (near-earth) is a classic Physics problem.
We assume that we have a symmetric top that can easily rotate about an axis containing its center of mass at a high angular velocity ω . If the top tilts at all from the vertical, so that its center of mass does not lie directly above the top's point of contact with the surface it's resting on, there will be a torque (single-axis) exerted on the top, which will act to change its angular momentum about a single axis. What will happen?
Solution
 Toggle Cloak
idsys
 System: 
 Cloak
idsys
The Spinning Top is a  with a large amount of .
 Toggle Cloak
idint
 Interactions: 
 Cloak
idint
 due to  and normal force from the surface the top is spinning on.
 Toggle Cloak
idmod
 Model: 
 Cloak
idmod
 .
 Toggle Cloak
idapp
 Approach: 
 Cloak
idapp
 Toggle Cloak
iddiag
  Diagrammatic Representation 
 Cloak
iddiag
If the top is perfectly upright, with its center of mass directly over the point of contact with the surface it's spinning on, then the torque (single-axis) about the point of contact is zero. The force due to gravity (near-earth) pulls directly downward, and the vector r between the point of contact and the center of mass points directly upward, so r X F = 0 .
Image Added
If the top tilts from the perfectly vertical orientation by an angle θ , then there will be a non-zero torque (single-axis).
Image Added
 Cloak
diag
diag
 Toggle Cloak
idmath
  Mathematical Representation 
 Cloak
idmath
The top is spinning about its axis with angular velocity ω The moment of inertia about its axis of rotation is I . For the case where the top has its axis perfectly vertical, the angular momentum is given by:
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|!559px-Spinning_top.jpg!|
|Spinning Top
from Wikimedia Commons: Image by User:Lacen|



{composition-setup}{composition-setup}

{excerpt}The rapidly spinning Symmetric Top exhibiting Precession under the force of [Gravity] is a classic Physics problem.{excerpt}

We assume that we have a symmetric top that can easily rotate about an axis containing its [center of mass] at a high [angular velocity] ω . If the top tilts at all from the vertical, so that its center of mass does not lie directly above the top's point of contact with the surface it's resting on, there will be a [torque (single-axis)] exerted on the top, which will act to change its [angular momentum about a single axis]. What will happen?

 



h4. Solution

{toggle-cloak:id=sys} *System:*  {cloak:id=sys}The Spinning Top is a [rigid body] with a large amount of [angular momentum about a single axis].{cloak}

{toggle-cloak:id=int} *Interactions:*  {cloak:id=int}[torque (single-axis)] due to [gravity] and normal force from the surface the top is spinning on.{cloak}

{toggle-cloak:id=mod} *Model:* {cloak:id=mod} [Angular Momentum and External Torque].{cloak}

{toggle-cloak:id=app} *Approach:*  

{cloak:id=app}

{toggle-cloak:id=diag} {color:red} *Diagrammatic Representation* {color}

{cloak:id=diag}

If the top is perfectly upright, with its [center of mass] directly over the point of contact with the surface it's spinning on, then the [torque (single-axis)] about the point of contact is zero. The force due to [gravity] pulls directly downward, and the vector *r* between the point of contact and the center of mass points directly upward, so *r X F = 0* .

|!Spinning Top Upright.PNG!|

If the top tilts from the perfectly vertical orientation by an angle θ , then there will be a non-zero torque.

|!Spinning Top Tipped.PNG!|


{cloak:diag}

{toggle-cloak:id=math} {color:red} *Mathematical Representation* {color}

{cloak:id=math}

The top is spinning about its axis with [angular velocity] ω The [moment of inertia] about its axis of rotation is *I* . For the case where the top has its axis perfectly vertical, the angular momentum is given by:

{latex}\begin{large}\[ \vec{L} = I \vec{\omega} \]\end{large}{latex}


The 
...
 direction 
...
 associated 
...
 with 
...
 both 
...
 the 
...
angular 
...
 velocity
...
 and 
...
 the 
...
 angular 
...
 momentum 
...
 is 
...
 directly 
...
 upwards.
...
If 
...
 the 
...
 axis 
...
 of 
...
 the 
...
 top 
...
 is 
...
 tipped 
...
 from 
...
 the 
...
 vertical 
...
 by 
...
 an 
...
 angle 
...
 θ , 
...
 then 
...
 the 
...
 situation 
...
 is 
...
 different. 
...
 The 
...
 angular 
...
 velocity 
...
 and 
...
 the 
...
 angular 
...
 momentum 
...
 both 
...
 point 
...
 along 
...
 this 
...
 angle 
...
 θ . 
...
 We 
...
 define 
...
 the 
...
 vector 
...
r
...
 that 
...
 runs 
...
 from 
...
 the 
...
 point 
...
 of 
...
 contact 
...
 (between 
...
 the 
...
 top 
...
 and 
...
 the 
...
 surface 
...
 it's 
...
 spinning 
...
 on) 
...
 and 
...
 the 
...
center 
...
 of 
...
 mass
...
 of 
...
 the 
...
 top. 
...
 The 
...
 force 
...
 of 
...
 gravity (near-earth), Fg , pulls on the center of mass, and this exerts a torque (single-axis)
...
 on 
...
 the 
...
 top. 
...
 The 
...
 torque, 
...
 τ , 
...
 is 
...
 given 
...
 by 
...
 the 
...
cross 
...
 product
...
 between 
...
r
...
 and 
...
F
...
g :
 Latex
{~}* :

{latex}\begin{large}\[ \vec{\tau} = \vec{r} \times \vec{F_{\rm g}} \]\end{large}{latex}

where

{latex}
where
 Latex
\begin{large}\[ F_{\rm g} = mg \]\end{large}{latex}


and, 
...
 as 
...
 the 
...
 gravitational 
...
 force, 
...
 points 
...
 straight 
...
 downwards.
Image Added
The magnitude of the cross product is given by
 Latex


|!Spinning Top Precession.PNG!|

The magnitude of the cross product is given by


{latex}\begin{large}\[ \mid \vec{r} \times \vec{F_{\rm g}} \mid = rmg\; sin(\theta) \]\end{large}{latex}


The 
...
 direction 
...
 associated 
...
 with 
...
 this 
...
 torque 
...
 is 
...
 horizontal 
...
 and 
...
 perpendicular 
...
 to 
...
 the 
...
 angular 
...
 momentum 
...
 vector 
...
L
...
. 
...
 As 
...
 a 
...
 result, 
...
 the 
...
 torque, 
...
 which 
...
 causes 
...
 a 
...
 change 
...
 in 
...
 the 
...
 angular 
...
 momentum 
...
 vector, 
...
 does 
...
 not 
...
 cause 
...
 a 
...
 change 
...
 in 
...
 the 
...
 magnitude 
...
 of 
...
 the 
...
 angular 
...
 momentum, 
...
 but 
...
 only 
...
 in 
...
 its 
...
 direction. 
...
 The 
...
 angular 
...
 momentum 
...
 vector 
...
 remains 
...
 tipped 
...
 at 
...
 the 
...
 angle 
...
 θ with 
...
 respect 
...
 to 
...
 the 
...
 vertical, 
...
 but 
...
 it 
...
 begins 
...
 to 
...
 rotate 
...
 around 
...
 the 
...
 vertical 
...
 axis 
...
 in 
...
 a 
...
 counter-clockwise 
...
 direction 
...
 with 
...
 an angular velocity Ω . This motion is called precession.
This new rotational motion ought to contribute to the angular momentum of the system. in most cases of interest and practical importance, however, the small addition to the angular momentum this causes is negligible compared to the angular momentum of the top rotating at angular velocity ω , and so it is usually ignored in the gyroscopic approximation, which holds that
 Latex
 angular velovitu Ω . This motion is called *precession*.


This new rotational motion ought to contribute to the angular momentum of the system. in most cases of interest and practical importance, however, the small addition to the angular momentum this causes is negligible compared to the angular momentum of the top rotating at angular velocity ω , and so it is usually ignored in the *gyroscopic approximation*, which holds that

{latex}\begin{large}\[ L = I \omega \]\end{large}{latex}


as 
...
 long 
...
 as 
 Latex
 

{latex}\begin{large}\[ \Omega \ll \omega \]\end{large}{latex}


In 
...
 that 
...
 case,the 
...
 change 
...
 in 
...
 angular 
...
 momentum 
...
 only 
...
 affects 
...
 the 
...
 horizontal 
...
 portion:
...
and 
...
 the 
...
 change 
...
 is
 Latex



{latex}\begin{large} \[  \rightleft| \frac{\Delta \vec{L}}{\Delta t} \leftright| = L \Omega sin(\theta) = rmg \; sin(\theta) \]\end{large}
and the precession angular velocity is 
 Latex
{latex} 

and the precession angular velocity is 

{latex}\begin{large} \[ \Omega = \frac{rmg}{L} = \frac{rmg}{I\omega}  \]\end{large}{latex}

{cloak:math}
{cloak:app}



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