# Question #93dc2

##### 1 Answer

Consider a solid sphere of radius

We need to calculate Moment of Inertia of this sphere about its diameter along

Consider a solid cylinder of thickness

We know that moment of inertia for a solid cylinder is given by the expression

#I=1/2MR^2#

#dI=1/2dmcdot r^2# ......(1)

#dm=ρcdotπr^2dx#

Substitute this value in equation (1)

#dI=1/2ρπr^4dx#

Writing

#dI=1/2ρπ(R^2–x^2)^2dx#

Integrating between limits

#I=1/2ρπint_(-R)^R(R^2–x^2)^2dx#

#=>I=1/2ρπint_(-R)^R(R^4–2R^2x^2+x^4)dx#

#=>I=1/2ρπ|R^4x–2R^2x^3/3+x^5/5|_(-R)^R#

#=>I=1/2ρπ|(R^4xxR–2R^2xxR^3/3+R^5/5)-(R^4(-R)–2R^2(-R)^3/3+(-R)^5/5)|#

#=>I=1/2ρπR^5(1-2/3+1/5+1-2/3+1/5)#

#=>I=1/2ρπR^5((15-10+3+15-10+3)/15)#

#=>I=1/2ρπ16/15R^5#

Density of the sphere is given by

Substituting in above we get