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(A) Let the tension in the cord be $T$. At the instant of release,the objects $A$ and $B$ are at rest,so their acceleration is zero. For object $A$,th

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$\frac{mg}{\sqrt{2}}$

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(A) Let the tension in the cord be $T$. At the instant of release,the objects $A$ and $B$ are at rest,so their acceleration is zero. For object $A$,the forces acting are gravity ($mg$ downwards) and tension ($T$ along the cord). The component of gravity along the cord is $mg \cos 45^{\circ}$. Since the object is constrained to move on the ring,the net force along the tangent to the ring must be considered. For object $A$,the component of gravity along the tangent is $mg \sin 45^{\circ}$. For object $B$,the component of gravity along the tangent is $mg \cos 45^{\circ}$. Since the cord is inextensible,both objects must have the same acceleration $a$ along the tangent. For $A$: $mg \sin 45^{\circ} - T = ma$ For $B$: $T - mg \cos 45^{\circ} = ma$ Adding the two equations: $mg(\sin 45^{\circ} - \cos 45^{\circ}) = 2ma$. Since $\sin 45^{\circ} = \cos 45^{\circ}$,$a = 0$. Substituting $a=0$ into the second equation: $T = mg \cos 45^{\circ} = \frac{mg}{\sqrt{2}}$.

Objects $A$ and $B$,each of mass $m$,are connected by a light inextensible cord. They are constrained to move on a frictionless ring in a vertical plane as shown in the figure. The objects are released from rest at the positions shown. The tension in the cord just after release will be

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