Geometry of the setup

Let the circular hoop have center at the origin, radius $R$.

• Bottom point of hoop: $(0,-R)$
• Top point of hoop: $(0,+R)$

The spring is attached at the bottom of the hoop, and its natural length is $R$.

Initial state

The bead is released from the top of the hoop with negligible speed.

At the top point, the distance from bottom point to top point is the diameter: $l_i = 2R$

Since the natural length is $R$, initial extension is $x_i = l_i - R = 2R - R = R$

So initial spring potential energy is $U_{s,i} = \frac12 kR^2$

Initial gravitational potential energy (taking bottom point as reference $y=0$ if convenient) corresponds to height difference $2R$ above bottom, so $U_{g,i} = 2mgR$

Initial kinetic energy: $K_i = 0$

Position when spring length becomes $R$

The bead lies on the hoop and is also at distance $R$ from the bottom attachment point.

So the bead is at the intersection of two circles:

• circle centered at hoop center, radius $R$
• circle centered at bottom point, radius $R$

Let the hoop center be $(0,0)$ and bottom point be $(0,-R)$.

Then bead coordinates satisfy $x^2+y^2=R^2$ and $x^2+(y+R)^2=R^2$

Subtracting, $y^2-(y+R)^2=0$ $y^2-(y^2+2Ry+R^2)=0$ $-2Ry-R^2=0$ $y=-\frac{R}{2}$

So the bead is at height $-R/2$ relative to the center. Since the top point was at $+R$, the vertical drop from top is $\Delta h = R-\left(-\frac{R}{2}\right)=\frac{3R}{2}$

Hence loss in gravitational potential energy is $\Delta U_g = mg\cdot \frac{3R}{2}$

At this position, spring length equals its natural length $R$, so spring potential energy is zero: $U_{s,f}=0$

Apply conservation of mechanical energy

Since there is no friction, $K_i+U_{g,i}+U_{s,i}=K_f+U_{g,f}+U_{s,f}$

Equivalently, the final kinetic energy equals the decrease in total potential energy: $\frac12 mv^2 = mg\left(\frac{3R}{2}\right)+\frac12 kR^2$

Multiply by 2: $mv^2 = 3mgR + kR^2$

Therefore, $v^2 = 3gR + \frac{kR^2}{m}$

So, $\boxed{v=\sqrt{3Rg+\frac{kR^2}{m}}}$

Option matching

This matches Option B.

Comparison with stored answer

Stored correct answer is B, which agrees with the derived result.