Final answer:
To balance the moments and make sure that the maximum moment within span AB is equal to the moment at the support B, the roller support should be placed halfway between A and B.
Step-by-step explanation:
The question relates to a scenario involving static equilibrium, specifically determining where to place a roller support on a beam so that the maximum moment within the span is equal to the moment at the support. The principles here revolve around the basic concepts of statics in structural engineering or physics, where one must balance moments and forces to ensure equilibrium.
To achieve the condition where the maximum moment within span AB is equivalent to the moment at support B, we need to derive the moments considering the weight distribution and reactions due to supports. Since this isn't a calculation-focused platform, it's essential to know that the correct placement of the roller support B would be to create symmetry in moments around the midpoint of the span. In static beam problems, this typically marks where bending moments reach their peak.
Given the options, the most likely placement would be halfway between A and B (option a), since it would result in a symmetrical distribution of moments about the center of the span. Placing the roller support at one-third or one-quarter of the distance would not equate the maximum moment within the beam to the moment at support B. Placing it at B would effectively create a cantilever beam, which has its maximum moment at the fixed support, rendering the condition unsatisfiable.