Page History

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

|!500px-Lissajous_curve_3to4.svgLissajous Figure.PNG!|
|Lissajous Figure
from Wikimedia Commons: Image by Peter D Reid|



{composition-setup}{composition-setup}

{excerpt:hidden=true}Image generated by a Pendulum with two natural Frequencies.{excerpt}



h4. Solution

{toggle-cloak:id=sys} *System:*  {cloak:id=sys}Flywheel as [rigid body] rotating about a fixed point under constant Torque.{cloak}

{toggle-cloak:id=int} *Interactions:*  {cloak:id=int}The fixed axis keeps the Flywheel from Accelerating. The Externally applied Torque.{cloak}

{toggle-cloak:id=mod} *Model:* {cloak:id=mod} Rotational Motion and Constant Torque.{cloak}

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

{cloak:id=app}

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

{cloak:id=diag}

It is important to sketch the situation and to define linear and rotational coordinate axes.

!Accelerating Flywheel 01.PNG!

{cloak:diag}

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

{cloak:id=math}

The force is supplied by a belt around the smaller wheel of radius *r* (in a 19th century factory, it would probably be a circular leather belt attached to the water wheels). This means that the direction the force is applied along is always tangential to the circumference of the wheel, and hence *Torque = r X F = rF*

{latex}\begin{large}\[ \vec{\tau} = \vec{r} X \vec{F} = rF = I_{\rm total} \alpha \]\end{large}{latex}

The Moment of Inertia of combined bodies about the same axis is simply the sum of the individual Moments of Inertia:

{latex}\begin{large}\[ I_{\rm total} = I_{\rm small} + I_{\rm large} \]\end{large}{latex}

The Moment of Inertia of a solid disc of radius *r* and mass *m* about an axis through the center and perpendicular to the plane of the disc is given by:

{latex}\begin{large}\[ I = \frac{1}{2}m r^2 \] \end{large}{latex}

So the Moment of Inertia of the complete flywheel is:

{latex}\begin{large}\[ I_{\rm total} = \frac{1}{2}(m r^2 + M R^2 ) \]\end{large}{latex}

The expression for the angular velocity and the angle as a function of time (for constant angular acceleration) is given in the *Laws of Change* section on the [Rotational Motion] page:



{latex}\begin{large}\[ \omega_{\rm f} = \omega_{\rm i} + \alpha (t_{\rm f} - t_{\rm i}) \] \end{large}{latex}

and

{latex}\begin{large}\[ \theta_{\rm f} = \theta_{\rm i} + \omega_{\rm i} ( t_{\rm f} - t_{\rm i} ) + \frac{1}{2} \alpha ( t_{\rm f} - t_{\rm i} )^2  \]\end{large}{latex}

We assume that at the start, *t{~}i{~} = 0* , we have both position and angular velocity equal to zero. The above expressions then simplify to:

{latex}\begin{large}\[ \omega_{\rm f} = \alpha t_{\rm f}\]\end{large}{latex}
and
{latex}\begin{large}\[ \theta_{\rm f} = \frac{1}{2} \alpha {t_{\rm f}}^2 \]\end{large}{latex}
where
{latex}\begin{large}\[ \alpha = \frac{rF}{I_{\rm total}} = \frac{2rF}{mr^2 + MR^2 }\]\end{large}{latex}

{cloak:math}
{cloak:app}



{td}
{tr}
{table}
{live-template:RELATE license}