Find the increase of moment of inertia of a steel sphere of diameter 10 cm about it's diameter when it's temperature is raised from 15C to...

We'll write the momentof inertia as a function of
temperature.

The radius of sphere is r0, at 0 degrees
Celsius. The mass of sphere is m.

The moment of inertia of
the sphere about the diameter
is:

(2/5)*m*rt^2

When the
temperature rises from 0C to tC, the nwe radius is:

rt =
r0(1+a*t)

The moment of inertia
is:

It = (2/5)*m*rt^2

It =
(2/5)*m*[r0(1+a*t)]^2

We'll expand the
square:

It = (2/5)*m*r0^2*(1 + 2a*t +
a^2*t^2)

We'll neglect the quantity a^2*t^2. We'll remove
the brackets:

It = (2/5)*m*r0^2 +
(4/5)*m*r0^2*a*t

It = I0 +
I0*2*a*t

We'll calculate the moment of inertia at t = 65
C

I65 = I0(1 + 2*a*65)

We'll
calculate the moment of inertia at t = 15 C

I15 = I0(1 +
2*a*15)

I65/I15 = 1 +
100*a

I65 = I15(1 + 100*a)

I65
- I15 = 100*a*I15

I15 =
2*4*5^5*7.7*pi/5*3

I15 =
8*7.7*625pi/3

I65 - I15 =
8*7.7*625pi*0.0012/3

I65 - I15 = 48.40 gm
cm^2