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 

 ^\x?pdxdydz 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, 0, &amp;lt;, we know that the polar element of 

 volume is r 2 sin 6drd6dtf&amp;gt;, and the z of any point is rcos 6. Hence 



I I jz*p dxdydz=\\ Lr 2 cos 2 0r 2 sin 6 dr d6 d(p, 



where the limits for r are and a, the limits for 6 are and ir, and the 

 limits for &amp;lt;p are and 2?r. 



a 5 a 2 



The value of the integral is p ^irM , where M is the mass of the 



5 5 



sphere, =f Trpa 3 . 



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

 2 = fa 2 . 



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

 an ellipsoid # 2 / 2 -|-^ 2 /& 2 -M 2 /c 2 =l, p being constant, we change the variables 

 by putting #=, 3/ = ^j z=c& and then we require the value of 



through a range of values given by the inequality 2 + 77 2 + 2 ;|&amp;gt;l. 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 -^irpcfibc. Hence, 

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



respectively. 



t Examples. 



1. Prove that a uniform rod, of mass ra and length 2&amp;lt;x, 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 



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 Jrafr 2 and %ma z . 



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 

 sides. 



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

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



