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A pendulum comprising a string of length L and a sphere swings in the vertical plane. The string hits a peg located a distance d below the point of suspension (see attached)

A) show that if the sphere is released from a height below that of the peg, it will return to this height after striking the peg.

B) Show that if the pendulum is released from the horizontal position (theta = 90 degrees) and is to swing in a complete circle centered on the peg, then theminimum value of d must be 3L/5.

I've come up with the following equations that I'm trying to relate to show the height after striking the peg is the same as before.

The initial height is L - Lcos[the], the height after striking the peg is L - d - (L - d)cos[psi].

mgh(initial) = (1/2)mv^2(bottom of the arc)

mgh(final) = (1/2)mv^2(bottom of the arc).

I'm not sure where to go from here.

Any suggestions would be greatly appreciated.

Thanks.

A) show that if the sphere is released from a height below that of the peg, it will return to this height after striking the peg.

B) Show that if the pendulum is released from the horizontal position (theta = 90 degrees) and is to swing in a complete circle centered on the peg, then theminimum value of d must be 3L/5.

I've come up with the following equations that I'm trying to relate to show the height after striking the peg is the same as before.

The initial height is L - Lcos[the], the height after striking the peg is L - d - (L - d)cos[psi].

mgh(initial) = (1/2)mv^2(bottom of the arc)

mgh(final) = (1/2)mv^2(bottom of the arc).

I'm not sure where to go from here.

Any suggestions would be greatly appreciated.

Thanks.