Final answer:
To find the value of 'x' in the given equation for relative velocity of a small sleeve sliding along a rotating rod, conservation of angular momentum is applied, considering there are no external torques on the system.
Step-by-step explanation:
A student is trying to find the value of x in the expression given for the relative velocity v' of a small sleeve sliding along a rotating rod, which is v' = ω₀l/√(1+xm/M). The rotation occurs in a horizontal plane about a stationary vertical axis passing through one end of the rod. To find x, we need to apply the principle of conservation of angular momentum since no external torques are acting on the system.
Initially, the angular momentum of the system is given by L₀ = I₀ω₀, where I₀ is the moment of inertia of the rod with respect to the axis of rotation and ω₀ is the initial angular velocity. As the sleeve moves to the other end of the rod, the new angular velocity ω and new moment of inertia I must satisfy L = Iω. Using the given information and solving for x, we can obtain the final value that satisfies the given conditions.
In this particular situation, we can ignore the specific details of the rod and sleeve's mass and length since we are provided with a general expression for v'. The aim is to apply the conservation law to relate the initial and final states of the system and solve for the unknown x.