40 



STRENGTH OF MATERIALS 



(22) /, + /. = /! + /.; 



that is to say, the sum of the moments of inertia with respect to any 

 two rectangular axes in the plane of the section is constant. 



(F) The numerical value of the moment of inertia is expressed as 

 the fourth power of a unit of length. Therefore the quantity - is 



the square of a length called the radius of 

 gyration, and will be denoted by /. The 

 radius of gyration is thus defined by the 

 equation 



(23) 



FlG 24 and is the average value of the distances 



of all the infinitesimal elements of area 

 from the axis with respect to which / is taken. 



The meaning to be attached to the radius of }j i tin- 



total area of the figure was concentrated in a single point at a distance 

 t from the axis, the moment of inertia of this single particle al"ut 

 this axis would be equal to the given moment of inertia. 



Problem 44. Find the moment of inertia of the rectangle in Problem 86 about 



its base, and also the corresponding radius of gyration. 



:**!, . = A. -+l'r- 



Problem 45. Find the moment of inertia of 

 the above rectangle about a gravity axis inclined 

 at an angle of 30 to its base. 



Problem 46. Find the moment of inertia of a 

 rectangular strip, such as that shown in 1 

 about a gravity axis parallel to its base. 



Problem 47. Prove that the moment of inertia 

 of a T-shape, such as that shown in Fig. 25, about 

 a gravity axis parallel to the base is given by the 

 expression 



b 



6 



Problem 48. Find the polar moment of inertia and radius of gyration of a circle 



of diameter d about an axis through its center. 



