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τ = I . α

where:

I : Moment of inertia (kg . m2)

α : Angular acceleration (rad / s2)

The moment of inertia I depends on the shape of the object and its distribution of pointmasses that build up the object. The moment of inertia can be calculated from:

I = Summation [point masses . (r2)].

where:

r : Radius measured from the axis of rotation.

E = 1/2 . I . ω2

where:

E : Energy (J)

I : Moment of Inertia (kg . m2)

ω : Angular velocity (rad / s)

P =

with:

P : Power (Watt)

ω : Angular speed (rad / s)

We explain the relevance of the formula τ = I . α by looking at the analogy of
the formula F = m . a, which we know from linear motion theory (Newton's Second Law).
The moment of inertia is, thus, explained as the ability to resist a change in *rotational*
motion. This means that a rotating object will stay rotating at the same rotational
speed as long as there are no working moments on the object.

The formula for the moment of inertia I highlights that it depends on the distribution of point masses in a body, relative to the axis of rotation. The greater the distance of point masses from the axis of rotation, the greater its distribution to the moment of inertia. The moment of inertia, clearly, depends on shape and the position of the axis of rotation.

The following table gives the moment of inertia for some common bodies:
Object |
Moment of inertia |

Massive cylinder, cylinder axis |
1/2 . m . R^{2} |

Thin rod, axis midway |
1/12 . m . l^{2} |

Thin disk, traverse axis through centre |
1/4 . m . R^{2} |

The formula for rotational energy also looks familiar, because it is analogous to the formula for the kinetic energy of a linear moving object. The stored rotational energy of an object depends on both the moment of inertia and the angular velocity.

The power is given by P = E / T =

I = (1/12) . m . L

We now start rotating this rod by applying a torque of 400 Nm. How long will it take for this rod to reach a speed of 500 rpm (revolutions per minute)?

τ = I . α

400 = 83,3α

α = 4.8 rad / s

ω = α . t

2π.500 / 60 = 4.8 t

100π/6 = 4.8t

t = 10,9 seconds

E = 1/2 . I . ω

P =

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