Question 13.196: A small sphere B of mass m is attached to an inextensible co...

Velocity at Point C (before the cord is taut).

Conservation of energy from B to C :

Velocity at C (after the cord becomes taut).

Linear momentum perpendicular to the cord is conserved:

\begin{aligned}& \theta=45^{\circ} \\& \overset{+}{\nwarrow} -m \nu_{C} \sin \theta=m \nu_{C}^{\prime} \\& \nu_{C}^{\prime}=(\sqrt{2 \sqrt{2}})\left\lgroup\frac{\sqrt{2}}{2}\right\rgroup \sqrt{g a} \\& \nu_{C}^{\prime}=\sqrt{\sqrt{2} g a}=2^{\frac{1}{4}} \sqrt{g a}\end{aligned}

Note: The weight of the sphere is a non-impulsive force.

Velocity at C :

Datum:

\begin{aligned}T_{C}+V_{C} & =T_{C^{\prime}}+V_{C^{\prime}} \\\frac{1}{2} m\left(\nu_{C}^{\prime}\right)^{2}+0 & =\frac{1}{2} m\left(\nu_{C}^{\prime}\right)^{2}+0 \\\nu_{C}^{\prime} & =\nu_{C^{\prime}}^{\prime}\end{aligned}

\underline{C^{\prime}} to \underline{C^{\prime \prime}} (conservation of energy)

\begin{aligned}& T_{C^{\prime}}=\frac{1}{2} m\left(\nu_{C^{\prime}}^{\prime}\right)^{2} \\& T_{C^{\prime}}=\frac{1}{2} m\left(2^{1 / 4} \sqrt{g a}\right)^{2} \\& T_{C^{\prime}}=\frac{\sqrt{2}}{2} m g a\end{aligned}

Datum:

\begin{aligned}T_{C^{\prime}}+V_{C^{\prime}} & =T_{C^{\prime \prime}}+V_{C^{\prime \prime}} \\V_{C^{\prime}} & =0 \\T_{C^{\prime \prime}} & =0 \\V_{C^{\prime \prime}} & =m g h \\\frac{\sqrt{2}}{2} m g a+0 & =0+m g h \\h & =\frac{\sqrt{2}}{2} a\end{aligned}

h=0.707 \mathrm{\ a}\blacktriangleleft