A uniform rod of mass m and length lo is rotating with a constant angular speed ω about a vertical axis passing through its point of suspension. The moment of inertia of the rod about the axis of rotation if it makes an angle θ to the vertical (axis$$ of $$rotation) is

Question

A uniform rod of mass m and length lo is rotating with a constant angular speed ω about a vertical axis passing through its point of suspension. The moment of inertia of the rod about the axis of rotation if it makes an angle θ to the vertical (axis$$ of $$rotation) is

Infinity Learn · Accepted Answer

We can observe each and every element of rod in rotating with different radius about the axis of rotationTake an elementary mass dm of the rod.dm $$=$$ $$ml0dlThe$$ moment of inertia of the elementary mass is given as dl $$=$$ $$(dm)r2The$$ moment of interita of the rod $$=$$  I $$=$$ ∫dI  ⇒I $$=$$ ∫$$r2dmSubstituting$$ r $$=$$ $$1$$ $$sin$$ θ and dm $$=$$ $$ml0$$.dl, we obtainI $$=$$ ∫$$(l2sin2$$θ$$)ml0dI$$ $$=$$ $$msin2$$θ$$l0$$∫$$0l0l2dl$$ $$=$$ $$ml023l0sin3$$θ⇒I $$=$$ $$ml02sin2$$θ$$3$$

A uniform rod of mass m and length l_o is rotating with a constant angular speed $ω$ about a vertical axis passing through its point of suspension. The moment of inertia of the rod about the axis of rotation if it makes an angle $θ$ to the vertical (axis of rotation) is

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A uniform rod of mass m and length lo is rotating with a constant angular speed ω about a vertical axis passing through its point of suspension. The moment of inertia of the rod about the axis of rotation if it makes an angle θ to the vertical (axis of rotation) is

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A uniform rod of mass m and length l_o is rotating with a constant angular speed $ω$ about a vertical axis passing through its point of suspension. The moment of inertia of the rod about the axis of rotation if it makes an angle $θ$ to the vertical (axis of rotation) is