Final answer:
The rotational kinetic energy of the grindstone is computed using the moment of inertia and angular velocity. Frictional work is equated to the initial kinetic energy via the work-energy theorem. We also account for the torque from friction to determine the angular deceleration and the total number of turns before stoppage.
Step-by-step explanation:
To solve for the rotational kinetic energy of the grindstone, we utilize the formula KE_rot = (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity. For a cylindrical grindstone, the moment of inertia, I, is given by (1/2)mr^2, where m is the mass and r is the radius. The angular velocity in radians per second can be found from the given rotation rate in rev/min by the conversion ω = (angular speed in rev/min) * (2π rad/rev) * (1 min/60 s). After calculating the kinetic energy, we can use the work-energy theorem to find the number of turns the grindstone makes before it stops.
To do this, we calculate the work done by friction, which is equal to the frictional force times the distance over which it acts. The kinetic friction force is given by f_k = μ_k * N, where μ_k is the coefficient of kinetic friction and N is the normal force, which, for a knife pressed against the grindstone, equals the force pressing the knife. For a circular path, the distance covered in one turn is the circumference of the grindstone, and the total distance is the circumference times the number of turns.
By equating the work done by friction to the initial kinetic energy, we can solve for the number of turns. We need to take into account that the force due to friction causes a torque that will decelerate the grindstone, resulting in an angular deceleration. Using the kinematic equations for rotational motion, we can relate the angular deceleration to the total number of turns before the grindstone comes to a stop.