Question

5) [8 pts] A friend of yours works in a bar as a bartender, and you decide to improve his life by making his job easier. In particular, he has to take empty beer mugs from the bar and put them in the sink (see the diagram below) - and of course, since students come in after exams for a drink, there are a lot of dirty beer mugs to move. You realize that he could send them sliding down the bar and they would land in the sink if they had the right initial speed. You wonder if you can find a single speed (so that he can easily learn to slide them correctly) for different conditions: in particular, if the bar is clean (larger coefficient of kinetic friction between the mug and the bar), or if there is some beer spilt on it (lower coefficient of kinetic friction). The vertical distance from the bar to the surface of the water in the sink is h=0.65m; the horizontal distances from the edge of the bar to the front and back of the sink are dmin=1.2m and dmax=1.8m. The coefficients of kinetic friction are pk=0.25 for a clean bar, and Uk=0.15 for a bar with beer spilt on it. Your friend stands at a point along the bar which is 5.6m from the edge of the bar, and an empty beer mug has a mass of 0.325kg. a) Find the minimum horizontal speed the mug must have when it leaves the bar so that it lands in the sink (at the near edge). b) Using the speed you found in a), calculate the bartender's "launch speed" (ie, the initial sliding speed) for the high-friction case (Uk=0.25). c) Using the speed you found in b) as the initial sliding speed, analyse the low- friction case: find the speed the mug will have when it leaves the bar, and determine whether it lands in the sink (or hits the far wall and shatters, thus making your friend lose his job). d) [for one extra point) Your bartender friend loves your new system, but the next time you visit him you realize he's still walking over to the sink every time he has a shot glass (a glass that's smaller than a beer mug - it's mass is only 0.125 kg) to put there. Does the mass of the glass matter, or can he launch them at the same speed as the mugs without worry (assume the same coefficients of friction as for the beer mug)? dmax beer mug bar sink