Circular motion question

I understand elementary physics pretty well and can easily conceptualize the following but cannot understand part of the math. Someone starts swinging a weight overhead on a rope and, of course, as the angular velocity increases the radius tries to increase. This is witnessed as the orbit moving higher up until the rope becomes more perpendicular with the axis of rotation. I understand that as omega increases the increasing centrifugal force acts outward on the weight, normal to the axis of rotation (thus trying to increase the radius). What I cannot understand mathematically is what component of the force is acting in the upward direction that causes the weight orbit to move upwards. Can someone show this using the proper equations?

Thanks for any help
Bailey--

Answers and Replies

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Welcome to the forums!

If you look at the mass/string from the side and draw a free body diagram, you'll see it.

You will have gravity acting on the mass, downward.
You will have the centrifugal force acting straight outward.
And you will have the tension in the string acting in the direction the string is pointing.

If the mass is not rotating, the tension in the string entirely cancels the force of gravity.

If the mass is rotating, horizontal component of the force of the sting counters the centrifugal force, and vertical component of the string counters the gravity.

For sake of argument, assume that we have the string rotating steadily where there is a -45 degree angle with the horizontal. Assume you then speed the mass' rotation up a little, and look at what is happening before the mass moves any (up or down).

The tension in the string increases to keep it in a circle. Since the tension is acting in both horizontal and vertical directions, and both increase, you now have an unbalanced force on the mass in the +vert direction. That is the force which causes it to rise.

The reason it doesn't rise forever is because as it is rising, the vert. component of the tension is decreasing as the horizontal component is increasing. When the vert. component = gravity, the mass is again in a steady 'orbit'