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Using Newton's second law to relate *F*_{t} to the tangential
acceleration
*a*_{t} = *r* , where is the angular
acceleration:

*F*_{t} = *ma*_{t} = *mr*

= *mr*^{ 2}.

= | |||

= |
(m_{i}r_{i}^{2}).
| (9) |

**Definition:** Moment of Inertia of a rigid body

The **moment of inertia**, *I* , of a rigid body gives a measure
of the
amount of resistance a body has to changing its state of
rotational motion. Mathematically,

I = m_{i}r_{i}^{2}.
| (10) |

This allows us to rewrite Equation 8.9 as:

= I
| (11) |

**Note: **

- The complete set of dynamical equations needed to describe
the motion of a rigid body consists of the torque equation given
above, plus Newton's Second Law
**applied to the center of mass of the object**:=

where is the acceleration of the center of mass.*m* - The moment of inertia, like torque must be defined about a particular axis. It is different for different choices of axes.
- Extended objects can again be considered as a very large collection of much smaller masses glued together to which the definition of moment of inertia given above can be applied.
- Examples of Moments of Inertia of Extended Objects:

uniform hoop:*I*=*mr*^{ 2}

cylindrical shell*I*=*mr*^{ 2}

long thin rod (about middle)*I*=*mL*^{ 2}

long thin rod (about one end)*I*=*mL*^{ 2}

solid cylinder*I*=*mr*^{ 2}

solid sphere*I*=*mr*^{ 2} - The moment of inertia depends on how the mass is distributed about the axis. For a given total mass, the moment of inertia is greater if more mass is farther from the axis than if the same mass is distributed closer to the axis.

10/9/1997