Question

A Sphere is rolling on plane surface .what fraction of its total kinetic energy is rotational.

Answer 1

Answer:

Given:

Solid sphere is rolling on a plane surface.

To find:

What fraction of total Kinetic energy is involved in rotational motion ?

Calculation:

Let mass be m, velocity be v , radius of Gyration be k , and total radius be r ;

We can say that total Kinetic energy is :

$KE = \dfrac{1}{2} m {v}^{2} \{1 + \dfrac{ {k}^{2} }{ {r}^{2} } \}$

Putting tge known values of k for solid sphere , we get :

$= > KE = \dfrac{1}{2} m {v}^{2} \{1 + \dfrac{ 2 }{ 5 } \}$

$= > KE = \dfrac{1}{2} m {v}^{2} \{ \dfrac{ 7 }{ 5 } \}$

Out of which the Translation Kinetic Energy is :

$Tr. \: K E = \dfrac{1}{2} m {v}^{2}$

Therefore Rotational Kinetic energy will be

$Ro. \: K E = K E - Tr. \: KE$

$= > Ro. \: K E = \dfrac{1}{2} m {v}^{2} \{ \dfrac{ 7 }{ 5 } \} - \dfrac{1}{2} m {v}^{2}$

$= > Ro. \: K E = \dfrac{1}{2} m {v}^{2} \{ \dfrac{ 2 }{ 5 } \}$

Ratio will be :

$\boxed{ \bold{ \dfrac{Ro. \: K E}{ \: K E} = \dfrac{ (\frac{2}{5}) }{ (\frac{7}{5} )} = \dfrac{2}{7} }}$

Answer 2

$\tt{\huge{\orange {Hello!!! }}}$

QUESTION :

A Sphere is rolling on plane surface .

What fraction of its total kinetic energy is rotational.

SOLUTION :

We know that the total kinetic energy is equal to the sum of the translational kinetic energy and the rotational kinetic energy.

We have to find ratio of Rotational Kinetic Energy to it's total Kinetic Energy..

Total Kinetic Energy is equal to

=> [ 1 / 2 ] m × v ^2 + [ 1 / 2 ] × I × { r } ^ 2...........(1)

We also know that :

I = [ 1 / 2 ] m { r } ^ 2 ...........(2)

Substituting the value of I in Equation 1 , We finally obtain the final result :

Total Kinetic Energy = [ 3 / 4 ] m { v } ^ 2

Rotational Kinetic Energy = [ 1 / 2 ] × I × { r } ^ 2........(3)

Now Substituting Equation ( 2 ) in Equation 1, We obtain the rotational kinetic energy which is equal to

=> [ 1 / 4 ] × m × r ^ 4.........( 5 )

Hence ratio

=> [ { 3 / 4 } × m × v ^ 2 ] / [ { 1 / 4 } × m × r^4 ]

=> [ 3 { v } ^ 2 ] / [ r ^ 4 ]>>>>>>>{ A }