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