Ball Suspended from Ceiling (Uniform Circular Motion)

[Gotejjeken]

Homework Statement

A ball of mass M = 0.5kg is suspended from a string whose other end is attached to the ceiling. The ball travels in a horizontal circle of radius R = 1.5m at a constant speed of v = 2 m/s.

http://online.physics.uiuc.edu/cgi/courses/shell/common/showme.pl?courses/phys211/oldexams/exam1/fa07/fig25.gif

A. What is the magnitude of the net force Fnet on the mass?

B. If the mass of the ball were increased while the speed of the ball was kept the same, how would the angle theta change?

Homework Equations

Fnet = m * v²/r

The Attempt at a Solution

A. I used a Free Body Diagram here with Tension pointing diagonally in the first quadrant, and weight pointing down on the y-axis. I came up with these equations:

(Fnet)x: Tx = Mb * (v²/r)
(Fnet)y: Ty - W = 0

Using the first of these equations I was able to solve for Tension and find that the magnitude of the net force is 1.33N.

B. Here is where I am a little confused at how to approach the problem. The part specifically asks for a mathematical proof in order to be correct, however I am unable to think of how to come up with such a proof.

I was able to come up with a basic idea by using the (Fnet) equations from above and the Pythagorean Theorem. I solved for (Fnet)x and (Fnet)y, then set up a right triangle and found theta to be 74.81 degrees. Then I doubled the mass and again set up a triangle with the new (Fnet)x and (Fnet)y values and found theta to be 74.60 degrees.

While this leads me to believe that theta will stay the same when the mass is changed and the velocity is left the same, it is not the mathematical proof that is asked for. Could someone please point me in the right direction to such a proof?

 

[Delphi51]

I think it will be very helpful to solve the 2nd equation for T, then sub into the first. The mass cancels so you'll get a relationship between v and theta.

 

[Gotejjeken]

Ah, wow. Thank you, it's the little things that are so often overlooked . I solved for T as you suggested and subbed it into get:

v = sqrt(g * r * cot(theta))

I suppose the proof would then be the steps leading up to this conclusion and the fact that the result is independent of mass, so no matter what mass the ball is the angle will still remain the same if the velocity is kept constant.

 

[Delphi51]

Right!
Check that again - I'm getting tan where you have cot.

 

[Gotejjeken]

Doh! I was using the wrong angle, and thus got cot(theta) instead of tan(theta). Thanks for the help.

 

[Delphi51]

Most welcome! Thanks for the interesting problem.