A square plate is supported in four different ways (configurations (P) to (S) as shown in the figure). A couple moment $C$ is applied on the plate. Assume all the members to be rigid and mass-less, and all joints to be frictionless. All support links of the plate are identical.
The square plate can remain in equilibrium in its initial state for which one or more of the following support configurations?
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In frictionless mechanisms, only closed link chains can resist an applied couple; open chains cannot prevent rotation.
The plate is subjected to a \textit{pure couple} $C$. For equilibrium, the support system must be able to resist rotation without allowing any relative motion at joints. All joints are frictionless, so rotation can only be resisted if there is a closed support mechanism capable of transmitting a moment.
Step 1: Analyze Configuration (P).
Configuration (P) forms an open chain with two links meeting the plate at three pivot joints. Since the members are massless and joints are frictionless, no moment can be transferred. The plate cannot resist the applied couple.
Therefore, (P) does \textit{not} provide rotational equilibrium.
Step 2: Analyze Configuration (Q).
Configuration (Q) forms a closed triangular linkage. A closed linkage can transmit moment because internal forces develop in the members even though joints are frictionless.
Thus, configuration (Q) can hold the plate in rotational equilibrium.
Step 3: Analyze Configuration (R).
Configuration (R) also forms a closed kinematic chain. A closed chain supports a couple moment because it prevents relative rotation between supports.
Hence, configuration (R) can resist the applied couple.
Step 4: Analyze Configuration (S).
Configuration (S) again forms a closed structure with two links at the bottom connected by pivots. Closed linkages block rotation even without friction.
Therefore, configuration (S) can resist the applied moment.
Step 5: Final conclusion.
Only (P) is incapable of resisting a couple.
Configurations (Q), (R), and (S) keep the plate in equilibrium.
Final Answer: (B), (C), (D)
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