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Question
The radius of a metal sphere at room temperature T is R, and the coefficient of linear expansion of the metal is α. The sphere is heated a little by a temperature ∆T so that its new temperature is T + ∆T. The increase in the volume of the sphere is approximately ______.
Options
2πRα∆T
πR2α∆T
4πR3α∆T/3
4πR3α∆T
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Solution
The radius of a metal sphere at room temperature T is R, and the coefficient of linear expansion of the metal is α. The sphere is heated a little by a temperature ∆T so that its new temperature is T + ∆T. The increase in the volume of the sphere is approximately `underline(4πR^3α∆T)`.
Explanation:
Let the radius of the sphere is R. As the temperature increases the radius of the sphere increases as shown.
Original volume `V_0 = 4/3 piR^3`
Coefficient of linear expansion = α
∴ Coefficient of volume expansion = 3α
∴ `1/V (dV)/(dT)` = 3α
⇒ dV = 3Vαdt ≃ 4πR3α∆T
= Increase in the volume
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