The uniform bar, having a cross-sectional area of A and mass...

The uniform bar, having a cross-sectional area of $A$ and mass per unit length of $m$, is pinned at its center. If it is rotating in the horizontal plane at a constant angular rate of $\omega$, determine the average normal stress in the bar as a function of $x$

Key Concepts

Stress Distribution in Rotating Bodies

Stress distribution refers to the variation of internal forces within a material. In a rotating bar, the normal stress arises due to the tension required to provide the centripetal force for each portion of the bar. By considering the equilibrium of differential elements along the bar, one can derive how the internal force—and hence the normal stress—varies as a function of the distance from the pivot point.

Differential Element Analysis

Differential element analysis involves breaking down the continuous body into infinitesimally small segments to analyze the forces acting on each segment. For the rotating bar, this method is used to integrate the effects of centripetal forces along its length. The cumulative effect of these forces determines the net tensile force at any section of the rod, which is then related to the average normal stress by dividing by the cross-sectional area.

Rotational Dynamics

Rotational dynamics deals with the motion of bodies that rotate about an axis. In this context, the bar’s rotation with constant angular velocity means that each point on the bar is undergoing circular motion and experiences a radial (centripetal) acceleration. This concept is fundamental for understanding the forces and resulting stresses in a rotating object, as the required centripetal force is provided by the internal stresses within the material.

Centripetal Force

Centripetal force is the inward force required to keep an object moving in a circular path. In a rotating bar, each differential mass element experiences a centripetal force proportional to the square of the angular velocity and its distance from the axis of rotation. This force must be balanced by the internal tensile forces within the bar, which in turn relates to the stress distribution along the rod.

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