Page History

{excerpt:hidden=true}Changes in Angular Velocity when the Moment of Inertia is changed, but no torque applied.{excerpt}


{composition-setup}{composition-setup}
{table:border=1|cellpadding=8|cellspacing=0|frame=void|rules=cols}
{tr:valign=top}
{td:width=310|bgcolor=#F2F2F2}
{live-template:Left Column}
{td}
{td}

|!Yun_2009_National_sit_spin.jpg!|
|Yun Yea-Ji (KOR) at the 2009 South Korean Figure Skating Championships 
Photo courtesy [Wikimedia Commons|http://commons.wikimedia.org]|

Consider an ice skater performing a spin. The ice is very nearly a frictionless surface 

A skater spinning around has constant angular momentum, but can change his or her Moment of Interia  by changing body position. What happens to their rate of rotation when they do so?


{deck:id=bigdeck}
{card:label=Part A}

h3. Part A

Assume that the skater originally is standing straight up while spinning, with arms extended. He or she then draws his or her arms inwards tightly to the sides. How does the angular velocity change?

h4. Solution





{toggle-cloak:id=sysa1} *System:*  {cloak:id=sysa1}The skater is treated as a [rigid body] in two configurations -- one with the arms extended, the other with arms held close.{cloak}

{toggle-cloak:id=inta1} *Interactions:* {cloak:id=inta1}External influences -- none. The skater is moving on a frictionless surface.{cloak}

{toggle-cloak:id=moda1} *Model:*  {cloak:id=moda1}[Single-Axis Rotation of a Rigid Body]{cloak}

{toggle-cloak:id=appa1} *Approach:*  

{cloak:id=appa1}

{toggle-cloak:id=diaga1} {color:red} *Diagrammatic Representation* {color}

{cloak:id=diaga1}

Consider the skater intially as a massless vertical pole with the arms modeled as massless rods of length {*}L{~}i{~}{*} with a point mass *m* at each end. 

|!Skater Initial 02.PNG!|

After contracting the skater's arms, the two masses are each a distance {*}L{~}f{~}{*} from the body

|!Skater Final 02.PNG!|


{cloak:diaga1}

{toggle-cloak:id=matha1} {color:red} *Mathematical Representation* {color}

{cloak:id=matha1}


The definition of the Moment of Inertia is:

{latex}\begin{large}\[ I = \sum {m r^{2}} \] \end{large}{latex}

 by 


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

 

{latex}\begin{large}\[ L_{\rm i} = I_{\rm i} \omega_{\rm i} = 2 m d_{\rm i} \omega_{\rm i} \] \end{large}{latex}

 

{latex}\begin{large}\[ L_{\rm f} = I_{\rm f} \omega_{\rm f} = 2 m d_{\rm f} \omega_{\rm f} \] \end{large}{latex}

 


{latex}\begin{large}\[  d_{\rm i} \omega_{\rm i} =  d_{\rm f} \omega_{\rm f} \] \end{large}

{latex}

or

{latex}\begin{large} \[\omega_{\rm f} = \omega_{\rm i} \frac{d_{\rm i}}{d_{\rm f}} \] \end{large}{latex}

This assumption really is not correct – Energy is NOT conserved in this case, despite the absence of outside forces. But we'll proceed to see where this assuimption leads.

 torques.

\\


{card:Part A}
{card:label=Part B}

h3. Part B

Assuming the ball is released from rest, what is the speed of the ball's center of mass after it has moved 1.3 m along the ramp?

h4. Solution

Once again, we solve the problem using two different methods.

{deck:id=littledeck2}
{card:label=Method 1}

h4. Method 1

{toggle-cloak:id=sysb1} *System, Interactions and Models:* {cloak:id=sysb1} As in Part A, plus [One-Dimensional Motion with Constant Acceleration|1-D Motion (Constant Acceleration)].{cloak}

{toggle-cloak:id=appb1} *Approach:* 
{cloak:id=appb1} 
After using one of the methods described in Part A to determine the acceleration, we can use kinematics to find the speed.  The most direct approach is to use:

{latex}\begin{large}\[ v_{x}^E_{\rm i} = \frac{1}{2} = v_{x,I_{\rm i}}^{2} + 2 a_{x} (x-x_{{\omega_{\rm i})}^2 \]\end{large}

{latex}

where we choose _x_~i~ = 0 m.  Solving gives:

{latex}\begin{large}\[ vE_{x} = \pm \sqrt{2a_{x}x rm f} = \pm \sqrt{\frac{2gx\sin\theta1}{1+\frac2}I_{\displaystylerm If}{\displaystyle mR^{2omega_{\rm f}}}}^2 \] \end{large}{latex}

we choose the plus sign, since the ball is translating

 in the + _x_ direction.  Since the y-velocity is zero, the total speed of the ball's center of mass is:

{latex}\begin{large}\[ v = \mbox{3.0 m/s} \] \end{large}{latex}
{cloak}

{card:Method 1}
{card:label=Method 2}

h4. Method 2

{toggle-cloak:id=sysb2} *System:*  {cloak:id=sysb2}The ball plus the earth and ramp. {cloak}

{toggle-cloak:id=intb2} *Interactions:*  {cloak:id=intb2}There are internal gravity, normal and friction forces.  Gravity is [conservative|work#nonconservative], while the normal force and friction are non-conservative.{cloak}

{toggle-cloak:id=modb2} *Model:*  {cloak:id=modb2}[Mechanical Energy and Non-Conservative Work].{cloak}

{toggle-cloak:id=appb2} *Approach:*  

{cloak:id=appb2}
Another way to solve the problem is to use energy.  It turns out that in this problem, the mechanical energy of the ball will be constant.  This assertion requires justification.  The forces present in the system are gravity, normal force and friction.  Gravity is a conservative force.  The normal force is non-conservative, but it does no work because it is perpendicular to the motion of the object.  Friction, however, is both non-conservative and directed anti-parallel to the motion of the ball, and so it should clearly do work.  The reason we can assume the energy is constant is the problem's statement that the ball rolls without slipping.  This means that the friction is static rather than kinetic.  Kinetic friction converts mechanical energy into thermal energy and so it is not appropriate to use conservation of mechanical energy when kinetic friction is present.  The work done by static friction, however, does not convert mechanical energy into thermal energy.  Instead, the static friction acts to divert some of the lost potential energy into rotational kinetic energy (rather than simply translational kinetic energy).  Thus, all of the energy remains in a mechanical form.

With this realization, we can write the equation of mechanical energy conservation in the form:

{latex}\begin{large}\[ K_{\rm f} + K_{\rm rot,f} + U_{\rm g,f} = K_{\rm i} + K_{\omega_{\rm f}}^2 = {\omega_{\rm i}}^2 \frac{d_{\rm i}}{d_{\rm f}} \]\end{large}

The above equation is NOT correct.

\begin{large}\[\Delta E = E_{\rm f} - E_{\rm {\rm rot,i} + U_{\rm g,i} \]\end{large}

{latex}

If we select _h_ = 0 at the point of release of the ball, then by the time the ball has moved a distance _x_ along the ramp, it has reached a height:

{latex} \begin{large}\[\Delta hE = - x \sin\theta\]\end{large}{latex}

Substituting zeros and appropriate expressions into the conservation of energy formula gives:

{latex}\begin{large}\[ \frac{1}{2}mv_{f}^{2}+ \frac{1}{2}I\omega_{f}^{2} - mgx\sin\theta = 0 \] \end{large}{latex}

Finally, the assumption that the ball is rolling without slipping implies the relationship:

{latex}\begin{large}\[ \omega_{f} R = v_{f}\]\end{large}{latex}

so:

{latex}\begin{large} \[ v_{f} = \sqrt{\frac{2gx\sin\theta}{1+\frac{\displaystyle I}{\displaystyle mR^{2}}}} \] \end{large}{latex}

{tip}The same result as obtained with method 1.{tip}

{info}Looking back at the equations of Part A, method 1 you can see that the friction force is:

{latex}\[ F_{f} = I\alpha /R \] {latex}

We know the value of α from Part A.  Thus, using the fact that the work done by friction when the ball moves a distance _x_ down the ramp is _W_ = -- _xF_~f~ we can write:

{latex}\[ W = - \frac{xI}{R^{2}} \frac{g\sin\theta}{1+\frac{\displaystyle I}{\displaystyle mR^{2}}} \]{latex}

Now, using our result for _v_~f~, we can write this as:

{latex}\[ W = - \frac{1}{2}I \frac{v_{f}^{2}}{R^{2}} = - \frac{1}{2}I\omega_{f}^{2} = - K_{\rm rot} \]{latex}

Thus, the work done by friction is exactly equal to the rotational kinetic energy acquired by the ball.{info}
{cloak}
{card:Method 2}
{deck:littledeck2}
{card:Part B}
{deck:bigdeck}

{td}
{tr}
{table}
{live-template:RELATE licenseE_{\rm i}(\frac{I_{\rm i}}{I_{\rm f}} - 1) \]\end{large}