merry go round.10 ft radius.8 oz steel ball resting @ perimeter.Groove in floor to center. MGR spinning 1 rev/sec. How much impulse force to push ball inside groove to 0 velocity as it passes center?
First, we need to be clear about what forces are acting when an object is in circular motion. In this case, there are no vertical forces other than the force of gravity keeping the roundabout, or merry-go-round, and the ball on the ground. The only horizontal force, neglecting any form of friction, is the centripetal force pulling the ball to the centre. This force depends on the speed of rotation and is mv^2/r, where v is the linear tangential speed of the ball or mw^2r where w is the angular speed, and r is the radius and m is the mass of the ball. The angular speed is 360 degrees per second (1 revolution per second) or 2(pi) radians per second. The actual speed of the ball is 2(pi)r feet per second, about 62.83ft/s or 42.84 mph. The speed is constant but the velocity varies because it's changing direction as a result of the centripetal force.
Since s, the sector arc, is r times the sector angle ø, the tangential speed is rw. So rw=v and v^2=r^2w^2 and v^2/r=rw^2. Hence we can express the force in two ways, and force is mass times acceleration. Now, this is where we have to be careful. The only force is the centripetal force, so what is centrifugal force? After all, we've all experienced an outward push whenever we move in a circle. Surely the ball experiences such an outward force? Well, this is an illusion, because there's only one force. The apparent centrifugal force is only an equal and opposite reaction to the centripetal force. It's sometimes called reactive centrifugal force. It's actually the inertial resistance of an object to being constrained to move in a circle. The object really wants to stay at rest or continue moving at the same velocity (that is, same speed and direction), but if the object is constrained from doing so by, say, the walls of the rotating body, the inertial resistance appears to be a force pressing the object against the walls. In a vehicle, for example, executing a turn, a passenger may fell pushed against the side of the vehicle, while what's really being experienced is that the passenger is trying to move straight ahead, but can't do so because of constraints imposed by the side of the vehicle.
So, since the ball is not constrained by a wall, it will naturally be flung out tangentially as the merry-go-round starts to rotate. If a force is applied it must be equal to mw^2r=0.5*4(pi)^2*10=197.4ft-lbs and this force has to be continuously applied as the machine rotates, otherwise the ball will simply be ejected. The alternative is a barrier or wall that moves with the merry-go-round. The groove will not affect this, because the groove is rotating too and will move "out of the way" as the ball moves tangentially unless restrained by the continuously applied force. There's no radial velocity so applying a radial impulse will just give the ball some brief change of momentum towards the centre, but as the system is rotating it will appear to an observer watching the merry-go-round that the ball is curving in towards the centre, the so-called Coriolis effect. The rotational speed of the ball will change because it has less distance to travel in one revolution. Think about stopping the system and giving the ball a brief push towards the centre. It will continue in a straight line along its groove directly to the centre having achieved some constant velocity as a result of the impulse. Now consider the system in motion. From the point of view of an observer riding with the ball, it will appear to move at constant speed to the middle. The centripetal force is taking care of the rotational velocity independent of this radial motion, So, in my view, we have two independent things happening here and an impulse is always going to increase the ball's speed to a constant, so that it never has a zero value when it reaches the centre.
I hope you appreciate that the above explanation is my understanding of the mechanisms at work, and I don't speak as an authority on the subject!
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