The radius of a metal sphere at room temperature T is R, and the coefficient of linear expansion of the metal is `alpha`. The sphere is heated a little by a temperature `DeltaT` so that its new temperature is `(T + DeltaT)`. The increases in the volume of the sphere is approximately

Options

A. `2pi Ralpha Delta T`
B. `pi R^(2)alpha Delta T`
C. `4pi R^(3)alpha Delta T//3`
D. `4pi R^(3)alpha Delta T`

Answer

D

Explanation

Let the radius of the sphere is R. As the temperature increases radius of the sphere increases as shown.
Original volume, `V_(o)=4/3piR^(3)`
Coefficient of linear expansion = `alpha`
`therefore` Cofficient of volume expansion = `3alpha`
`therefore1/V(dV)/(dT)=3 alpharArrdV=3ValphadT~~4piR^(3)alphaDeltaT` = increase in the volume