Max Tangential Speed of Mass Before Cord Breaks

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The discussion focuses on calculating the maximum tangential speed of a mass attached to a string before the cord breaks, which is determined to be 4.33 m/s given a maximum tension of 113.1 N. In the second part, the mass is analyzed while moving in a vertical circle at a speed of 6.69 m/s, where the tension at the top of the circle is calculated to be 2.45×10^2 N. The user initially applied the wrong formulas for part b, neglecting the effect of gravity on tension during vertical motion. A correction is provided, emphasizing the need to account for gravitational force in the tension equation. Proper application of the formulas leads to the correct tension calculation.
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A mass M of 2.47 kg is attached to the end of a string whose length is 0.410 m. The mass slides without friction on a horizontal surface as indicated in the diagram.

a)If the string can withstand a maximum tension of 113.1 N, what is the maximum tangential speed the ball can have before the cord breaks?
Answ: 4.33 m/s

b)The mass, with a new string, is whirled in a vertical circle of the same radius about a fixed point. Find the magnitude of the tension when the mass is at the top if its speed at the top is 6.69 m/s.
Answ: 2.45×102 N


For part a) i used the formulaa 1) F=ma 2)a=rw^2 3) V=rw to get the right answer but for part b) I used 1)V=rw 2) a=rw^2 3)T=ma but I am not getting the right answer. Am I doing something wrong?
 
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Diagram please.

BJ
 
b) the motion is vertical, so you must take gravity into account.
 
In the second case, gravity is also acting.
So, mv^2/r=mg+T.
 
T = m ((v^2/r)-g)
 
Kindly see the attached pdf. My attempt to solve it, is in it. I'm wondering if my solution is right. My idea is this: At any point of time, the ball may be assumed to be at an incline which is at an angle of θ(kindly see both the pics in the pdf file). The value of θ will continuously change and so will the value of friction. I'm not able to figure out, why my solution is wrong, if it is wrong .
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