answer:** To calculate the total kinetic energy of the system, we need to consider both the translational kinetic energy and the rotational kinetic energy. The translational kinetic energy is given by: {eq}rm KE_{trans} = frac{1}{2}mv^2 {/eq} where m is the mass of the sphere and v is its linear velocity. The rotational kinetic energy is given by: {eq}rm KE_{rot} = frac{1}{2}Iomega^2 {/eq} where I is the moment of inertia of the sphere and omega is its angular velocity. The total kinetic energy of the system is then: {eq}rm KE_{total} = KE_{trans} + KE_{rot} {/eq} Plugging in the given values, we get: {eq}rm KE_{trans} = frac{1}{2}(0.01 kg)(2pi times 0.1 m times 2 rps)^2 = 0.008 J {/eq} {eq}rm KE_{rot} = frac{1}{2}left(frac{2}{5}(0.01 kg)(0.005 m)^2right)(10 rps)^2 = 0.0001 J {/eq} {eq}rm KE_{total} = 0.008 J + 0.0001 J = 0.0081 J {/eq} Therefore, the total kinetic energy of the system is 8.1 mJ.

answer:The divergence of a vector field mathbf{F}(x, y, z) = f(x, y, z)uvec{i} + g(x, y, z)uvec{j} + h(x, y, z)uvec{k} is given by nabla cdot mathbf{F} = frac{partial f}{partial x} + frac{partial g}{partial y} + frac{partial h}{partial z}. Let's compute the partial derivatives: 1. frac{partial f}{partial x} = 0 since f does not depend on x. 2. frac{partial g}{partial y} = frac{1}{z left(1 - frac{y^2}{z^2}right)} from the derivative of arctanh. 3. frac{partial h}{partial z} = frac{1}{z} from the derivative of ln. Now, we sum these partial derivatives to find the divergence: nabla cdot mathbf{F} = 0 + frac{1}{z left(1 - frac{y^2}{z^2}right)} + frac{1}{z} Simplifying, we obtain: nabla cdot mathbf{F} = frac{1}{z left(1 - frac{y^2}{z^2}right)} + frac{1}{z} This result is correct, as it shows the sum of the rates of change of each component with respect to the corresponding coordinate.

answer:The first step is to find the derivative of f(x): f'(x) = -81x^2 At x = 5, we have: f'(5) = -81 cdot 5^2 = -2025 The function value at x = 5 is: f(5) = -27 cdot 5^3 = -3375 The inverse function's series is given by: f^{-1}(x) = a + frac{f'(a)}{f''(a)}(x - f(a)) Using x = -3375 and a = 5, we find the coefficient a in the first-order expansion: a = f^{-1}(f(a)) = f^{-1}(-3375) However, since we don't know f^{-1}(x) explicitly, we can express a as: a = -frac{f(a)}{f'(a)} = frac{-3375}{-2025} = 5 Now, the first-order series of the inverse function is: f^{-1}(x) = 5 + frac{1}{-2025}(x - (-3375)) f^{-1}(x) = 5 - frac{1}{2025}(x + 3375) f^{-1}(x) = frac{3375 - x}{2025} - 5 Thus, the first-order series of the inverse function around x = 5 is: frac{3375 - x}{2025} - 5

answer:In the given sentence, "La hermana de mi hermana es mi _______________," the subject is "La hermana de mi hermana" (my sister's sister), and the verb is "es" (is). The missing word after "es" is a noun that describes the relationship between my sister's sister and me. Since my sister's sister is my sibling, the correct word to fill in the blank is "hermana" (sister). Therefore, the complete sentence is "La hermana de mi hermana es mi hermana" (My sister's sister is my sister). My sister's sister is my sister.