A solid sphere is in rolling motion. In rolling motion a body possesses translational kinetic energy `(K_(t))` as well as rotational kinetic energy `(K_(r ))` simultaneously. The ratio `K_(t):(K_(t)+K_(r ))` for the sphere is

Options

A. `2:5`
B. `7:10`
C. `10:7`
D. `5:7`

Answer

D

Explanation

`("Linear kinetic energy")/("Total kinetic energy")=("Linear kinetic energy")/("Rotational K.E. + Linear K.E.")`
`(K_(t))/(K_(t)+K_(r))=((1)/(2)mv^(2))/((1)/(2)Iomega^(2)+(1)/(2)mv^(2))`
`=((1)/(2)mv^(2))/((1)/(2)xx(2)/(5)mr^(2)xx(v^(2))/(r^(2))+(1)/(2)mv^(2))[becauseI=(2)/(5)mr^(2)andv=romega]`
`=((1)/(2)mv^(2))/((1)/(5)mv^(2)+(1)/(2)mv^(2))`
`=((1)/(2))/((1)/(5)+(1)/(2))`
`=((1)/(2))/((7)/(10))=(5)/(7)`