## To answer the question, we need to calculate the work required to change the rotational rate of the particles glued to the rod. However, the information provided is incomplete.

Here are the steps you can take to approach this problem:

1. Calculate the moment of inertia (I) of the system. The moment of inertia depends on the distribution of mass and the axis of rotation. Given that the particles are glued to the rod and can rotate around a perpendicular axis through point O at one end, you need to treat the system as a combination of point masses.

2. Determine the radius of the particles from the axis of rotation. This information is needed to calculate the moment of inertia. However, the radius is not provided in the given information. You may need to obtain this information from the problem statement or make an assumption based on the given data.

3. Once you have determined the moment of inertia, you can calculate the work done. The work-energy theorem states that the work done on an object equals the change in its kinetic energy. Since the kinetic energy of a rotating system is given by K = 0.5 * I * Ï‰^2, where Ï‰ is the angular velocity, you can use this equation to calculate the change in kinetic energy and hence the work done.

4. For parts (a), (b), and (c) of the question, substitute the given angular velocities into the equation and calculate the corresponding work done.

5. Finally, to answer part (d) of the question about the slope of the plot of the assembly's kinetic energy versus the square of its rotational rate, you need to plot the kinetic energy (in joules) on the y-axis and the square of the rotational rate (in radians-squared per second squared) on the x-axis. Calculate the slope of this plot by taking the change in kinetic energy divided by the change in the squared rotational rate.

Remember to complete the missing information (such as the radius of the particles) before proceeding with the calculations.