Maximizing Tension in Vertical Circular Motion with Work-Energy Theorem

[Juntao]

A child is playing with a ball on a string. The ball has mass 0.065 kg, the string has radius 1.2 m and assume that the string is massless. The child swings the ball on a string in a vertical circle. Halfway up the circle the speed of the ball is measured to be 8.7 m/s. What is the maximum tension in the string?

I know that a=V^2/R

F=MV^2/R

And since I'm working in the work-energy chapter this week, I think maybe I could apply the work energy theorem to this problem, but not sure how or even how to start this problem.

So help greatly appreciated.

 

[HallsofIvy]

You need to be able to calculate the velocity at each point. In order to use F= ma= mV²/R. Try using "conservation" of energy. You know the velocity "halfway up the circle" so you can calculate the kinetic energy. Calculate the potential energy since this is a vertical circle. Add to find the total energy. Subtracting off the potential energy at every other point let's you find the kinetic energy and so the velocity (squared) at each point.

 

[Juntao]

Um, what?
Sorry, but I don't follow what you mean by this "Subtracting off the potential energy at every other point let's you find the kinetic energy and so the velocity (squared) at each point."

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Anyhow, can someone check this work logic please?

I figured out the kinetic energy at that point,
Ke=1/5*m*v^2
= .5*.065kg*(8.7m/s)^2
= 2.46J

PE= m*g*h=> .065kg*9.8*1.2m =.765 J.

Total energy =3.225J.

So now I need velocity.
3.225J=.5*m*v^2
v=9.96 m/s

Now I know that F=mv^2/R
so F=.065kg*(9.96)^2/1.2m =>5.37N

So my max force or Tension rather is going to be 5.37N. Does that look right guys?

 

[kylemadigan]

hey m8, do you mean the circle is spinnning horizontally over his head or vertically next to him? i found out horizontally i think, but if its vertically spinning next to him let me know. i got like 4.47 Newtons, but that's if the ball is spinning above his head horizontally. lots of redundancy in my reply eh?

 

[HallsofIvy]

Kylemadigan: did you notice that the title of this thread was "Vertical Circular Motion help"? Also the original post said "The child swings the ball on a string in a vertical circle. "

Juntao: You did exactly what I said: except that you reasoned correctly that the greatest speed will be where the potential energy is 0- so you "subtracted off" 0! My point about "velocity (squared)" was that you once you got "3.225J=.5*m*v^2" you didn't really need to take the square root to get v: Since the formula for centripetal force also uses mv^2, you only need mv^2= 3.225/(.5)= 6.45.

Then F= mv^2/r= 6.45/1.2= 5.375 N just as you got.

 

[Juntao]

For some reason, the answer is not working out with the computer. Hmm...maybe the circle isn't uniform?

 

[jamesrc]

It's probably not working because at the position the maximum tension occurs, the tension of the string must support the centripetal force AND the weight of the object. Try adding the weight of the object to your answer.