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Mohr’s Circle for Mom. of Inertia Class Today • Print notes and examples • Mohr’s Circle for Moments of Inertia – Defined – Finding max and min values – Finding angle of rotation • When all possible moments and products of inertia for a shape are plotted on a graph, the resulting graph is a circle. • This graph makes it easy to find any moment or product of inertia for a shape that we want. All we have to do is use the geometrical properties of a circle. • Example Problems • Group Work Time 1 Mohr’s Circle for Mom. of Inertia • Ix and Iy are plotted on horizontal axis • Ixy is plotted on vertical axis • Plot point ‘X’ 1st (Ix , Ixy) • ‘Y’ will be on opposite side as (Iy ,-Ixy) (note signs of Ixy) • Now you have two points • Draw the circle through these points • Find any other points you need (usually Imin and Imax) 3 2 Finding Imin and Imax • Once the circle is drawn (see previous slide) locate the center point (shown here as O) • Form a right triangle by dropping a vertical line from point X to horizontal axis. • Use this triangle to find the radius of the circle (hypotenuse) and the angle shown as 2θ • Imin and Imax are simply the center plus or minus the radius 4 Finding Theta • It can be important to know the orientation (with respect to the x-y axis system) of a physical object for a given moment of inertia. • Rotating the object’s coordinate system through any angle θ will correspond to an angle of 2θ on Mohr’s Circle. • Theta (θ) is always measured: – the same rotational direction on both the object and the graph. – from point X on the graph and from the x-axis on the object. θp1 θp1 5 27_Mohr's Circle 1