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 `Delta T` so that its new temperature is `T+ Delta T`. The increase in the volume of the sphere is approximately

Options

A. `2pi R alpha 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_(0) = (4)/(3)pi R^(3)`
Coefficient of linear expansion `= alpha`
`:.` Coefficient of volume expansion `= 3alpha`
`:. (1)/(V)(dV)/(dT) = 3 alpha rArr dV = 3V alpha dt ~= 4pi R^(3) alpha Delta T`
= Increase in the volume