Wondering about Moment of inertia
So here it is,MOMENT OF INERTIA-From now on ,mark my words really carefully,I don’t want to frighten you but you know as dreadful as it sounds,so dreadful is its definition-It is defined as the summation of product of mass of each particle and the square of its distance from the axis of rotation.Told you,that really sounded horrible.
Life is already very complicated so let’s make it simple silly.Simply put,Moment of Inertia(I) of a body is just a number that tells us how easy it is ,to rotate that body about an axis and that’s all about it.Done.Got it?The lower this number,the easier it is to rotate the body.
Now how do we measure it?As a thumb rule,if the mass is concentrated away from the axis of rotation,the body has a higher moment of inertia and more difficult it is to rotate the body.And vice versa i.e. if the mass is concentrated near the axis of rotation,the lower the moment of inertia and it is easier to rotate this body.Just remember-mass away->higher MOI->difficult to rotate,mass near->lower MOI->easier to rotate.
Coming back to our question,why did the solid sphere win the race?Well,I guess you know the answer now.Yes you guessed it right!! All the bodies have same mass and hence same normal reaction force and hence the same friction acting on all of them.
But the solid sphere has most of that mass concentrated near its axis.So it has a lower moment of inertia and hence it rotates easily and wins the race.
Time for some technicalities
As we can see that the sphere is rotating about its centre,so all the forces passing through the centre do not provide any torque.This leaves us with friction as the only force that can provide torque.
Now using the formula,
Γ=Ια ,where Γ is torque, Ι is moment of inertia, α is angular acceleration
We get α =Γ/Ι i.e. α is inversely proportional to Ι for constant Γ.
Now Γ=fs x r,r is radius of the sphere which is same for all the four bodies.,fs is the static friction and it is also same for all(since fs=µmgcosθ which is same for all).Hence Γ is constant for all the four competitors. So lower Ι means higher α.And among the four competitors of our race only the solid sphere with least Ι has highest α.So it rotates the fastest and wins the race.So now we know why did solid sphere win the race and won many hearts!! 🙂