Sounds like an experiment! Actually, that doesn't seem to make much sense if you consider that most weeds are weeds because they are so resilient. They are prolific propagators and/or they require little care. I.e. they will grow in compacted soils with little nutrients. I would doubt that most broadleaf weeds wood be killed off by such harsh treatment unless it was done for a long period of time in which case the grass would also suffer. Secondly, cutting the grass short allows dormant weed seeds on the soil surface access to light causing them to germinate making the whole discussion moot.