A ball is swung in a circular path

AI Thread Summary
A 0.5 kg ball is swung in a circular path from an 80 cm rope at a 40-degree angle, with the speed calculated to be approximately 2.056 m/s and kinetic energy at this angle determined to be about 1.06 J. For the scenario where the kinetic energy drops to 50% of its initial value, the discussion focuses on finding the new angle theta. The approach involves using equilibrium equations and trigonometric identities, with suggestions made to solve the resulting quadratic equation after substituting cosine for theta. The conversation emphasizes the importance of analytical methods over trial and error in solving the problem. Overall, the thread highlights the application of physics principles and trigonometry in determining the ball's behavior in circular motion.
234jazzy2
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Homework Statement


A 0.5 kg ball is swung in a circular path from a 80 cm long rope, attached to a vertical pole.
A. What is the speed of the ball theta (between the pole and rope) is 40 deg?
B. What is the KE of the ball at theta = 40 deg?
C. The ball slows down and the KE drops to 50% of the value in (B). What is the new value of theta?

Homework Equations


F = ma
centripetal acceleration = V^2/r

The Attempt at a Solution



A.
Fy = 0
Tcos(theta) = mg
Fx = mv^2/r
Tsin(theta) = mv^2/r - > V = sqrt(g*r*tan(theta)) = 2.056 m/s
B. KE = 0.5 *m*v^2 = 1.057151179
C. I get all the have to new velocity but i don't know how to get the angle... I need some pointers.

Also, i am not sure if this is the right approach.

Thanks
 
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Hello jazzy, :welcome:

Looks like the right approach. A few remarks: KE = 1.06 J (don't forget the units and don't give many more digits than the given variables have -- but if the first digit is a one, then give one more).

For C, you have the same equilibrium equation (##\ v^2 = g\, r \tan\theta\ ##), only now v is given and ##\theta## has to be determined. Your problem is then the goniometric equation when you put in ##r = L \sin\theta## (L is the length of the rope).

If you have no way to solve this, perhaps you are supposed to find the answer with trial and error ?
 
Yea, i get suck at the trig. And, it's definitely not trial and error. Trying different reference frame to see if I can get rid of a trig.
 
234jazzy2 said:
Yea, i get suck at the trig. And, it's definitely not trial and error. Trying different reference frame to see if I can get rid of a trig.
What trig equation do you get? Something like sin(θ)tan(θ)=value? There is an analytic way to solve that.
 
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(Sin^2(theta))/( cos(theta)) = some number. I tried using some trig identities but nothing seemed to work. As I write this, I think I could've solved it, because that also equals (1 - cos^ 2(theta))/ cos(theta) = something and set x = cos(theta) and sove the quadratic. But that will give two answers... I'll solve it later. But if you have any othersuggestions, please let me know.
 
234jazzy2 said:
set x = cos(theta) and solve the quadratic.
That is the method I had in mind.
 
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