216-218] BODIES OF SPECIAL FORMS. 231 



IV. Sphere. Let a be the radius of the sphere, p its density (supposed 

 uniform), and let it be referred to a system of rectangular axes with 

 origin at the centre. Then we require the values of such integrals as 

 Nffipdxdydz taken through the volume. From symmetry it is clear that 

 the integrals obtained from this by putting y or z for x will have the same 

 value as the one written. 



Now using polar coordinates r, 6, <f>, we know that the polar element of 

 volume is r 2 sin 6drdBd^>^ and the z of any point is rcos 6. Hence 



I I fz 2 pdxdydz = I 1 1 p r 2 cosW sin 0drd6d(p, 



where the limits for r are and , the limits for 6 are and TT, and the 

 limits for <p are and 2n. 



a 5 a 2 



The value of the integral is p %2ir= M - , where M is the mass of the 



5 5 



sphere, =irpa 3 . 



Thus, if k is the radius of gyration of the sphere about a diameter, 

 2 =fa 2 . 



V. Ellipsoid. To find the value of ^x^pdxdydz through the volume of 

 an ellipsoid # 2 /a 2 +?/ 2 /& 2 +,s 2 /c 2 =l, p being constant, we change the variables 

 by putting #=|, y=brj, z=c, and then we require the value of 



through a range of values given by the inequality t?+i? + i?^>\. The 

 integration may therefore be regarded as taken through the volume of a 

 sphere of radius unity, and thus by No. IV. the result is ^irpcPbc. Hence, 

 if M is the mass of the ellipsoid, the moments of inertia about the axes are 



iJ/^+c 2 ), lM(c*+a*), 4J/(a 2 + &2), 

 respectively. 



^ Examples. 



1. Prove that a uniform rod, of mass m and length 2a, has as momental 

 equivalent three particles, one of mass f m at its middle point, and one of 

 mass %m at each of its ends. 



2. Prove that the moments of inertia of a uniform rectangular disc, of 

 mass m and of sides 2a, 26, about lines through its centre parallel to its edges 

 are \mb 2 and %ma?. 



3. Prove that the radius of gyration of a uniform circular disc about a 

 diameter is half the radius. 



4. Prove that the moments of inertia of a uniform elliptic disc, of mass 

 m and semiaxes a and 6, about these axes are %mb 2 and %ma\ 



5. Prove that a uniform triangular disc has as momental equivalent 

 three particles, each one-third of its mass, placed at the middle points of its 



6. Prove that the moment of inertia of a uniform cube about any axis 

 through its centre is raa 2 , where m is the mass and 2a the length of a side. 



