Can anyone help with this misconception.
Take a particle mass m moving in a vertical circle speed v.
At the lowest point applying F=ma we have T-mg = (mv^2)/r. Ok that's fine.
At the highest point we have T+mg =(mv^2)/r , this is where I am getting confused both the tension and weight act down so what is keeping the particle up? Is there another force at play? How do we quantify it?
Many thanks
Throw a ball straight up. What keeps it in the air? It is not “kept” in the air. It is constantly accelerating (g). As it goes up it loses kinetic energy and gains potential energy till it reaches a maximum height where initial kinetic energy equals potential energy.
Same thing happens with a ball constrained (by string or track) to move in a vertical circle. If you give it an initial tangential (horizantal) velocity it will just reach the top of the circle if the initial velocity is the same it would be if you threw it straight up to reach the same height. Centrifugal force doesn’t influence kinetic energy of ball because it’s perpendicular to the motion. It is assumed there is no friction.
If you throw it up faster than KE=PE, then tension in the string will keep it moving indefiniteley in a circle. If KE=PE, it will reach the top and then drop straight down.
If KE<PE, it will go partway around the circle and drop till constrained by the string. If it doesn't reach half-way-up, it will just swing back and forth like a play-ground swing.
What if it goes past half-way-up but not to top? Then it drops till constrained by string; in this case, how far will it go in the other direction after it starts swinging in an arc again? It depends on the energy lost when it hits string tension. If it goes almost to the top, it will drop straight down almost to the bottom at which point it sees string tension but has almost no tangential velocity to keep it going in an arc. How far it will go after hitting string (track) depends on the height where it hits track and component of tangential velocity when it hits track. Then KE+PE = constant determines how far it goes.
Whoops! I pissed in your ear and told you it was raining. Sorry, a little bit rusty on this.
If initial KE (at bottom) is less than mgR, PE half way up, ball will just oscillate: weight will always have a component pulling string taut. R= radius of circle of dia D.
If initial KE is greater than mgR, string will remain taut if centrifugal force is greater than the component of gravity in the direction of the string inward. For the string to remain taut at top, for ex, it must be going fast enough so that centrifugal force is equal to weight; in this case initial kinetic energy must equal PE at top plus KE of vcentrifugal.
This sets up the calculation:
1) KE at any point is KEinitial minus mgh. That gives tangential velocity at any point.
2) Calculate min tangential velocity at any point so that centrifugal force = component of weight in dir of string (perpendicular to track).
3) Ball will leave track before top when tangential velocity is less thn 2).
EDIT: If it's a cart on a track, and it has enough KE to take it to say 145deg before leaving track, where does it land? Calculate min velocity to keep it on track at 145deg. Then it becomes another classic problem: if I fire the cart with initial speed and direction off a cliff, where does it land?