A tornado can be idealized as a Rankine vortex with a core of diameter $30$ m$.$ The gauge pressure at a radius of $15$ m is $2000$ N$/$m$2($that is$,$ the absolute pressure is $2000$ N$/$m$2$ below atmospheric$).($a$)$ Show that the circulation around any circuit surrounding the core is $5485$ m$2/$s$.[$Hint: Apply the Bernoulli equation between infinity and the edge of the core.$]($b$)$ Such a tornado is moving at a linear speed of $25$ m$/$s relative to the ground. Find the time required for the gauge pressure to drop from $500$ to $2000$ N$/$m$2.$ Neglect compressibility effects and assume an air temperature of $25$C$.($Note that the tornado causes a sudden decrease of the local atmospheric pressure. The damage to structures is often caused by the resulting excess pressure on the inside of the walls, which can cause a house to explode.$)$