The nail of figure 5.23 P located at a distance d below the suspension point of
ID: 2184674 • Letter: T
Question
The nail of figure 5.23 P located at a distance d below the suspension point of the pendulum O. Show that d must be at least 0.6L for the small sphere of mass m of the pendulum can turn around in a circle with center spot.
The ball complete a full circle if anywhere in his career (except quizasa at point D) the tension T in the string is null. In other words, that the mass m would describe a full circle and must always remain taut rope.
Should it happen that at an arbitrary point C- in which Vc? 0- the voltage is zero, the string would not still attached to the ball, so this body will continue moving as freely and describe a parabolic trajectory. Therefore, the condition is that T> 0 to??180
Explanation / Answer
Conservation f energy before the nail:
m g L = 0.5 m vi^2
vi = (2gL)
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Conservation of angular mmentum:
Li = Lf
==> Ii i = If f
Ii = m L^2
i = vi/L = (2gL)/L = (2g/L)
If = m r^2
f = vf/r
Ii i = If f
==> (m L^2) ((2g/L)) = (m r^2) (vf/r)
==> (2gL^3) = (r) (vf)
==> r = ((2gL^3))/vf
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Conservation f energy after the nail:
m g (2r) = 0.5 m vf^2
vf = (4gr)
Therefore:
==> r = ((2gL^3))/((4gr))
==> r = (L^3/2r)
==> 2 r^3 = L^3
==> r = L/(32) = 0.7937 L
we have:
d = L - r = L - 0.7937 L = 0.206 L