question_answer 29)

                  The radius of a metal sphere at room temperature T is R, and the coefficient w linear expansion of die 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                 (a) \[2\,\pi \,R\,\alpha \,\Delta \,T\]                  (b) \[\pi \,{{R}^{2}}\,\alpha \,\Delta \,T\]                 (c) \[4\pi \,{{R}^{3}}\,\alpha \,\Delta \,T/3\]      (d) \[4\pi \,{{R}^{3}}\,\alpha \,\Delta \,T\]                

Answer:

                  (d) \[\Delta V=\gamma V\Delta T\]                 \[=\,3\,ax\,\frac{4}{3}\,\pi {{R}^{3}}\Delta T\]                 \[=4\pi \,{{R}^{3}}\,\propto \,\Delta T\]