Two conceptual questions regarding Newton's laws

[feveroffate]

1. Why is it that when you swing a ball around your head, the string or rope attached to the ball is always tilted down a little? That is, the rope isn't parallel to the floor, it is at a bit of angle at down.

2. The ball will continue to move in a circular path if the total force is (mv^2)/r. But what if the force becomes a little weaker than this (but is still directed toward the center of the circle)? In which direction would the ball travel? There is still a force on it, just a little less than (mv^2)/r.

 

[Delphi51]

Consider the vertical forces on the ball - when the string is horizontal.

 

[feveroffate]

Just gravity?

 

[Delphi51]

Yes. So the ball will accelerate downward. The horizontal ball-and-string is unstable. It will go down until there is an upward force to counteract gravity.

 

[feveroffate]

so an object will only move in a circular path if the force is equal to or greater than mv^2/r?

 

[Delphi51]

so an object will only move in a circular path if the force is equal to or greater than mv^2/r?

Okay, on to the second question. The condition Fc = mv^2/r is required for circular motion. If the force is larger or smaller, the path will no longer be circular. For example if a spacecraft in circular motion around the Earth fires retro rockets reducing v, the Fc (gravity) will then be greater than the new mv^2/r. The spacecraft will then be in an elliptical orbit.

 

[feveroffate]

so if the force were less than the original Fc required for circular motion, the ball would move in an elliptical orbit, but sort of inward?

and regarding the first question, is it possible to calculate the angle that the string is "down", if i had mass, radius, and angular velocity? F = m R w^2, but how would the angle be incorporated?

 

[Delphi51]

so if the force were less than the original Fc required for circular motion, the ball would move in an elliptical orbit, but sort of inward?

Outward. The starting point would be the perihelion; radius would increase until on the opposite side of the Earth (aphelion).

To calculate the angle, make a free body diagram for the ball and consider the vertical and horizontal forces separately. Vertically, the sum of the forces is zero because it does not accelerate. Horizontally, the force equals mv^2/r. These two equations - each involving the tension in the string - can be solved as a system to find the angle.

 

[feveroffate]

I'm a little confused. In the y direction, the force due to gravity is the only force? And in the x direction, F = mv^2/r. So where would the angle come into play?

 

[Delphi51]

The string exerts a force on the ball in both directions. Call the combined force T (short for tension). Use sine and cosine to get an expression for the vertical and horizontal parts of it. Usually the T can be found by solving the vertical part for T, then substituting that expression for the T in the horizontal part.