Final answer:
To find the velocity of the rolling sphere, we can use the conservation of energy. By calculating the translational and rotational kinetic energies of the sphere and summing them up, we can determine the total kinetic energy. Given the mass, radius, and translational velocity of the sphere, we can calculate its angular velocity, rotational inertia, and then find the total kinetic energy, which is 64 J.
Step-by-step explanation:
To find the velocity of a solid uniform sphere that rolls without slipping across a level surface, we can use the conservation of energy. The total kinetic energy of the rolling sphere is equal to the sum of its translational kinetic energy and rotational kinetic energy. The translational kinetic energy can be calculated using the formula T = 0.5 * m * v^2, and the rotational kinetic energy can be calculated using the formula K = 0.5 * I * w^2, where m is the mass of the sphere, v is its translational velocity, I is its rotational inertia (moment of inertia), and w is the angular velocity. Since the sphere is rolling without slipping, we can relate its translational and rotational quantities using the equation v = R * w, where R is the radius of the sphere.
Given that the mass of the sphere is 2.00 kg, the radius is 0.350 m, and the translational velocity is 4.00 m/s, we can first find the angular velocity w using the equation v = R * w. Rearranging the equation, we have w = v / R = 4.00 m/s / 0.350 m = 11.4 rad/s. Next, we can calculate the rotational inertia I of the sphere using the formula I = (2/5) * m * R^2, which gives us I = (2/5) * 2.00 kg * (0.350 m)^2 = 0.49 kg*m^2.
Now that we have all the required values, we can calculate the total kinetic energy of the rolling sphere using the formula K_total = T + K = 0.5 * m * v^2 + 0.5 * I * w^2 = 0.5 * 2.00 kg * (4.00 m/s)^2 + 0.5 * 0.49 kg*m^2 * (11.4 rad/s)^2 = 64 J. Therefore, the total kinetic energy of the rolling sphere is 64 J.