Hello everyone and welcome to Jamma Academy. In this example a cube of mass M lies on a ton table rotating at a constant angular velocity omega. We are required to determine the maximum angular speed before the cube sleeps outwards and we also have to explain why. In other words, if the angular speed of the ton table is slowly increased, at some instance the cube will actually fly outwards in a straight line. So the question is why is that the case? Now for you to be able to see exactly why, we have to start first by drawing our free body diagram. Now the cube is rotating in a circle and therefore it is performing circular motion. That means that it has a centripetal acceleration and by Newton's second law for a system to have an acceleration there must be a net force acting on that system. Under direction of the net force is the same as the direction of the acceleration of the object. That means that the net force acting on the cube is directed towards the center of the cube. So the question that should be in your mind is what then provides or what is the centripetal force in this particular instance? For you to see the answer to this question let's start first by doing our free body diagram. So the weight of the block is vertically downwards. There is a normal force acting upwards and the block has a tendency to slide to skip outwards. That means that there is a force preventing it from sliding outwards. Therefore that force must be acting inward. And so we have here a static friction force acting inward. So if we draw our free body diagram this is static friction. This is the weight and you have here the normal force. So by definition we know that the sum of all forces in the y direction this is equal to n minus mg that will be equal to 0. So n is equal to mg. The normal force acting on the block or on the cube is equal to the weight of the cube. By definition we know that f of s is less than or equal to mu s n. In other words f of s is less than or equal to mu s mg. Now let us sum the forces in the direction of the centripetal acceleration. We know that the summation of f is equal to m a r which would mean that f of s is equal to m a r. In other words f of s is equal to m r omega squared. Now there are two equations I would like for you to take a look at which are very important for you to understand why the cube will skip outwards after we reach a certain value of omega. Let me level this equation one and equation two. From equation one f of s has a maximum value that it can attain. f of s has a maximum value that it can attain and that maximum value is given by mu s mg. Now that means that if the centripetal force required to keep the cube moving in a circle exceeds this maximum value the cube will slide outwards. Now when you look at equation two as omega increases f of s also increases that means that as we continually increase or gradually increase the angular speed of the cube of the turn table the centripetal force required to keep the cube moving in a circle consequently will also increase. Hence if we continue to increase omega there will come a point in which the centripetal force required is greater than the maximum static friction. When that is the case there will not be a sufficient force to compare the cube to move in a circle at that particular point it will skip outwards in a straight line in accordance with Newton's first law of motion. So that means that f s max which is equal to mu s mg will be equal to m r omega max squared. The m's will cancel and the maximum angular velocity that we can have and the system will not skeet is given by this particular expression is given by this particular expression. Well thank you so much for your time I really do appreciate your time and commitment for this course and look forward to the next lesson. Bye bye.