In a nuclear reactor, heat is generated uniformly in the 5 -cm-diameter cylindrical uranium rods at a rate of \(2 \times 10^{8} \mathrm{~W} / \mathrm{m}^{3}\). If the length of the rods is \(1 \mathrm{~m}\), determine the rate of heat generation in each rod. Answer: \(393 \mathrm{~kW}\)

To find the rate of heat generation in each rod, we first need to calculate the volume of each rod. The rods have a cylindrical shape and we are given the diameter and the length. The formula for the volume of a cylinder is: \[V = \pi r^2 h\] where 'V' is the volume, 'r' is the radius, and 'h' is the height (length) of the cylinder. Since we have the diameter (5 cm) and length (1 m), we can find the radius by dividing the diameter by 2: \[r = \frac{d}{2} = \frac{5\mathrm{~cm}}{2} = 2.5\mathrm{~cm}\] Now, convert the radius and length to meters: \[r = 2.5\mathrm{~cm} \times \frac{1\mathrm{~m}}{100\mathrm{~cm}} = 0.025\mathrm{~m}\] \[h = 1\mathrm{~m}\] Now plug in the values for radius and height into the volume formula: \[V = \pi (0.025\mathrm{~m})^2 (1\mathrm{~m})\]

Now we have to calculate the actual volume of each cylindrical rod. Using the formula and the values we just found, we can find the volume: \[V = \pi (0.025\mathrm{~m})^2 (1\mathrm{~m}) = 0.001963\mathrm{~m}^3\]

We are given the heat generation rate per cubic meter, which is \(2 \times 10^{8} \mathrm{~W} / \mathrm{m}^{3}\). To find the rate of heat generation in each rod, we just need to multiply the rate in \(\mathrm{W} / \mathrm{m}^{3}\) by the volume of the rod: \[\textit{Rate of heat generation} = 2 \times 10^{8} \mathrm{~W} / \mathrm{m}^{3} \times 0.001963\mathrm{~m}^3 = 393000\mathrm{~W}\]

Lastly, we need to express our answer in kilowatts. To do this, divide the watts by 1000: \[\textit{Rate of heat generation} = \frac{393000\mathrm{~W}}{1000} = 393\mathrm{~kW}\] Hence, the rate of heat generation in each uranium rod is 393 kW.