A heavy particle is suspended by a string of length L. The particle is given a horizontal velocity vo. The string becomes slack at some angle and the particle proceeds on a parabola. Find the value of vo if the particle passes through the point of suspension.
When the string becomes slack the forces become equal
outward (centrifugal force) => mv^2/L
inward => mg cosø
mv^2/L = mg cosø
v= √Lgcosø
Energy is conserved – initial ke => mvo^2/2 = m(v)^2/2 + mgL(1+cosø)
=> vo^2 = m(Lgcosø) + 2mgL(1+cosø)
vo^2 = 3gLcosø + 2gl => 1
To find cosø-
L cosø = 1/2 gt^2 – vsinø t => 2
L sinø = v cosø t
t = L sinø/cosø v
t= L tanø/v => 3
replacing in 2
=> L cosø = L tan^2ø / 2 cosø – L sin^2ø/cosø
=> L cosø = L sin^2ø/ 2cos^3ø – 2L sin^2ø cos^2ø/ 2cos^3ø
=> 2L cos^4ø = L sin^2ø – 2L sin^2ø cos^2ø
=>2L cos^4ø + 2L sin^2ø cos^2ø = L sin^2ø
=> 2Lcos^2ø(cos^2ø + sin^2ø) = L sin^2ø
=> 2Lcos^2ø = Lsin^2ø
=> 2 = tan^2ø
=> tanø = √2
=> cosø = 1/√3
Replacing cosø in 1
=> vo^2 = 3gLcosø + 2gL
=> vo^2 = 3gL*1/√3 + 2gL
=> vo^2 = gl(√3+2)
=> vo = √(gl(√3+2))