Solve Ball on a String for Velocity w/ Mass, Length & Gravity

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AI Thread Summary
A problem involving a ball of mass M on a massless string of length L is presented, where the ball is projected with an initial horizontal velocity V and swings before hitting the nail. The key to solving the problem is determining the conditions under which the tension in the string becomes zero, allowing the ball to transition from circular to parabolic motion. Initial calculations yielded an angle of 65.53 degrees, but further review led to a revised angle of 54.7 degrees. The final expression for the projection velocity V is derived as V = √(3.24Lg). The discussion emphasizes the importance of thorough calculations and verification in solving physics problems.
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This isn't actually a homework problem, but I thought it would be relevant to post it here since it is a problem afterall, and I need some help at the solution. Thank you everyone!

Homework Statement



There is a ball mass M hanging on a massless string length L, from a nail in the wall. When the ball is projected with initial horizontal velocity V, it swings up, but then falls back down and hits the nail. Find V in terms of M L and g.

2. Requested Help

I would appreciate it if someone could see if my method is problematic, and if my answer is correct if you would want to work it out.

The Attempt at a Solution



For ball to leave a circle and transit to a parabolic motion, Tension in string must equal to zero at point of transition. Thereafter, the parabolic motion will coincide with the nail. I have 3 equations and 3 unknowns - velocity at moment of transition, time taken for parabolic flight, and angle from vertical which transition occurs.

Using the above, I got an angle of 65.53 deg
I got v^2 = gl cos(angle)
And I eliminated t.

Finally, after using conservation of energy, I arrive at projection velocity V = Sqrt(3.24Lg)
 
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Welcome to PF!

Hi melancholy2! Welcome to PF! :smile:

(have a square-root: √ and a degree: º and try using the X2 tag just above the Reply box :wink:)
melancholy2 said:
For ball to leave a circle and transit to a parabolic motion, Tension in string must equal to zero at point of transition. …

Yes, that's correct. :smile: But it's very difficult to check your final result without seeing your full calculations. :redface:
 
Hi everyone,

I've relooked at my workings and managed to come up with a new answer which I think is correct. Angle is now 54.7 degrees. Please see the attachment solution. Thanks!
 

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Hi melancholy2! :smile:

(I haven't actually checked the last 6 lines :redface:, but apart from that …)

Yes, that looks fine. :smile:

(except you could have saved yourself a little trouble if you'd noticed that lcosθ + lsin2θ/cosθ = l/cosθ :wink:)
 

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