Final answer:
The rotating flywheel with an initial kinetic energy of 1.0 x 10^6 J and subject to a frictional torque of 4.0 N×m would take 250 seconds to come to rest. The time was calculated using the work-energy principle and the motion equations for rotating bodies.
Step-by-step explanation:
The question is asking for the time it would take for a rotating flywheel with an initial kinetic energy of 1.0 x 106 J and an angular speed of 400 rad/s to come to rest when a frictional torque of 4.0 N×m is applied to it.
To calculate the time, we start with the work-energy principle, which states that the work done by the frictional torque will equal the change in kinetic energy of the flywheel. Since the flywheel is coming to rest, the final kinetic energy will be zero and the change in kinetic energy will be equal to the negative of the initial kinetic energy (-1.0 x 106 J). The work done by the frictional torque over an angle θ is given by W = τθ, where τ is the torque and θ is the angular displacement in radians.
First, we find the angular displacement needed to bring the flywheel to rest by using the equation W = τθ, where W is -1.0 x 106 J and τ is -4.0 N×m (negative because it opposes the rotation). This gives us the angular displacement θ required to stop the flywheel. Then, we use the equation θ = 0.5ω0t + 0.5αt2, where ω0 is the initial angular velocity (400 rad/s), α is the angular acceleration, and t is the time. Because the flywheel comes to a stop, ω = 0 rad/s, so we rearrange the equation to solve for t.
By doing this calculation, we found that t = 250 seconds. Therefore, the correct answer is (b) 250 seconds.