A bead of mass \( m \) can slide without friction on a fixed vertical loop of radius \( R \). Th...

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A bead of mass \( m \) can slide without friction on a fixed vertical loop of radius \( R \). The bead moves under the combined effect of gravity and a spring with spring constant \( k \) only. The spring is rigidly attached to the bottom of the
loop. Assume that the natural length of spring is zero. The bead is released from rest at \( \theta=0^{\circ} \) with non-zero but negligible speed to the bead. The gravitational acceleration is directed downward as shown in the figure. Now choose the correct option(s). (Neglect any type of friction between any contact) (given \( k R=m g / 2 \) )
(1) The speed \( v \) of the bead when \( \theta=90^{\circ} \) is \( \sqrt{3 g R} \).
(2) The speed \( v \) of the bead when \( \theta=90^{\circ} \) is \( \sqrt{5 g R} \).
(3) The magnitude of the force that ring exerts on the bead when \( \theta=90^{\circ} \) is \( 3 \mathrm{mg} \).
(4) The magnitude of the force that ring exerts on the bead when \( \theta=90^{\circ} \) is \( 5 \mathrm{mg} \).
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