Knowing that the collar is passing through the point D with a speed of $1.8 m/s,$ determine the speed of the collar when it passes through point C and B. Take stiffness of the spring, $K= 250N/m,$ Radius of the circular path $= 300 mm$ and distance $OA = 125 mm.$

We assume the spring is resolving freely about A. We also assume there is no loss of energy in the system.

Let BD be the ground level i.e. at which height $h = 0.$

Radius $= OC= OB = OD = 300mm=0.3m \\ OA = 125mm = 0.125m \\ \therefore AB=0.3-0.125=0.125 m $

Un-deformed length of the spring $= 0.175m.$

Stiffness of the spring $k = 250 N/m$

Let $v_b,v_c,v_d $ be the velocities of the collar at the positions B, C, D respectively.

When the collar is at B, h=0,

$KE_B=\dfrac 12 mv^2_a $ and $PE_B=0$

Extension in spring $ (x_b ) =0$

Spring energy $(E_s ) =0$

When the collar is at D, h=0,

$KE_D=\dfrac 12 mv_d^2 $ and $PE_D=0 \\ AD = OA + OD = 0.125 + 0.3 =0.425m$

Extension in spring $(x_d )= AD-AB = 0.425-0.175 = 0.25m $

Spring energy $= E_s = \dfrac 12 Kx_d^2$

Applying work energy principle to the collar at positions B and D,

$U_{B-D} = KE_D-KE_B \\ \therefore PE_B -PE_D -E_s= KE_D -KE_B \\ \therefore 0-0 -\dfrac 12 Kx_d^2 =\dfrac 12 mv_d^2 -\dfrac 12 mv_b^2 $

Multiplying by 2,

$-250 \times 0.25^2 = 1 \times 1.8^2 - 1 \times v_b^2 \\ \therefore v_b^2 =1.8^2 +250 \times 0.25^2 \\ \therefore v_b=4.3434 m/s $

When the collar is at $C, h=OC = 0.3m. \\ KE_c=\dfrac 12 mv_c^2 \text {and } PE_C=mgh$

In $\triangle $AOC, by Pythagoras theorem,

$$AC=\sqrt{OA^2+OC^2} $$

$=\sqrt{0.125^2+0.3^2} \\ =0.325 mm$

Extension in spring $(x_c ) = AC-AB \\ =0.325-0.175=0.15m$

Spring energy $= E_s =\dfrac 12 Kx_c^2$

Applying work energy principle to the collar at positions B and C,

$U_{B-C}=KE_C-KE_B \\ \therefore PE_B-PE_C -E_S =KE_C - KE_B \\ \therefore 0-mgh -\dfrac 12 Kx_c^2 =\dfrac 12 mv_c^2 -\dfrac 12 mv^2_b \\ \therefore -1\times 9.81\times0.3-0.5\times 250 \times 0.15^2=0.5\times1\times v_c^2 -0.5\times1 \times 4.3434^2 \\ \therefore -2.943-2.8125 +9.4326 =0.5 v_c^2 \\ v_c=2.7119 m/s $

Hence,

Speed of the collar when it passes through point $C = 2.7119 m/s$

Speed of the collar when it passes through point $B = 4.3434 m/s $