Final answer:
The total kinetic energy of a rolling disk can be calculated by summing the translational kinetic energy and the rotational kinetic energy. In this case, the mass and radius of the disk are given, as well as its angular velocity. By using the appropriate equations, the translational and rotational kinetic energies can be calculated and summed to find the total kinetic energy of the disk.
Step-by-step explanation:
The total kinetic energy of a rolling disk can be calculated by summing the translational kinetic energy and the rotational kinetic energy.
The translational kinetic energy is given by the equation KE_trans = (1/2)mv^2, where m is the mass of the disk and v is its linear velocity. In this case, m = 2 kg and v = rω, where r is the radius of the disk and ω is its angular velocity. Given that r = 0.5 m and ω = 5 rad/s, we can calculate v = rω = 0.5 m * 5 rad/s = 2.5 m/s. Plugging these values into the equation, we find KE_trans = (1/2)(2 kg)(2.5 m/s)^2 = 6.25 J.
The rotational kinetic energy is given by the equation KE_rot = (1/2)Iω^2, where I is the moment of inertia of the disk and ω is its angular velocity. The moment of inertia of a solid disk rotating about its axis is given by the equation I = (1/2)mr^2. Plugging the values of m and r into this equation, we find I = (1/2)(2 kg)(0.5 m)^2 = 0.25 kg·m^2. Plugging this value and ω = 5 rad/s into the equation for KE_rot, we get KE_rot = (1/2)(0.25 kg·m^2)(5 rad/s)^2 = 6.25 J.
Finally, we can find the total kinetic energy by summing the translational and rotational kinetic energies: KE_total = KE_trans + KE_rot = 6.25 J + 6.25 J = 12.5 J.
Therefore, the total kinetic energy of the rolling disk is 12.5 J.