Final answer:
The moment of inertia of a professional boxer's forearm is found using the relation between torque, force, lever arm, and angular acceleration. The forearm's rotational kinetic energy is calculated using the moment of inertia and angular velocity derived from linear velocity and distance from the elbow joint.
Step-by-step explanation:
The calculation of moment of inertia is a fundamental concept in physics that relates to the rotational motion of objects. In the case of the professional boxer, the muscle force, lever arm, and angular acceleration are provided to find the moment of inertia of the forearm. The torque applied by the muscle can be calculated using the formula τ = F × r, where τ is the torque, F is the force, and r is the lever arm. The angular acceleration (α) is directly proportional to the torque and inversely proportional to the moment of inertia (I), as described by Newton's second law for rotation: τ = I × α.
To solve for the moment of inertia in this scenario, we use the values given: force (F) of 2.00×10³ N, lever arm (r) of 3.00 cm (0.03 m), and angular acceleration (α) of 120 rad/s². The torque can be calculated as follows: τ = (2.00×10³ N) × (0.03 m) = 60 N·m. Then, rearrange the formula to find the moment of inertia: I = τ/α = 60 N·m / 120 rad/s² = 0.5 kg·m².
The rotational kinetic energy of the forearm can be calculated using the formula KE_rot = (1/2)Iω², where ω is the angular velocity. The linear velocity (v) and the distance (r) from the elbow joint allow us to find ω using the formula ω = v/r. With a linear velocity of 20.0 m/s and a distance of 0.480 m, ω is 41.67 rad/s. Plugging these values into the rotational kinetic energy formula gives KE_rot = (1/2)(0.500 kg·m²)(41.67 rad/s)² = 434.03 J.