Three links are hinged together to form a triangle A B C as...

Three links are hinged together to form a triangle $A B C$ as shown in the figure. At a certain instant, the point $A$ is moving towards the mid point of $B C$ with a velocity of $5 \mathrm{~m} / \mathrm{s}$ and $B$ is moving at a perpendicular direction to $A C$. The velocity of point $C$ is : (a) $5 \mathrm{~m} / \mathrm{s}$ (b) $10.4 \mathrm{~m} / \mathrm{s}$ (c) $10.8 \mathrm{~m} / \mathrm{s}$ (d) $1.8 \mathrm{~m} / \mathrm{s}$

Three links are hinged together to form a triangle $A B C$ as shown in the figure. At a certain instant, the point $A$ is moving towards the mid point of $B C$ with a velocity of $5 \mathrm{~m} / \mathrm{s}$ and $B$ is moving at a perpendicular direction to $A C$. The velocity of point $C$ is :
(a) $5 \mathrm{~m} / \mathrm{s}$
(b) $10.4 \mathrm{~m} / \mathrm{s}$
(c) $10.8 \mathrm{~m} / \mathrm{s}$
(d) $1.8 \mathrm{~m} / \mathrm{s}$

Key Concepts

Relative Velocity Analysis

This concept involves comparing the velocities of different points on a rigid body or system by considering their motion relative to one another. In mechanisms with interconnected parts, relative velocity analysis helps determine unknown speeds by applying vector addition and subtraction based on the constraints imposed by the rigid connections.

Instantaneous Center of Rotation

In planar kinematics, any rigid body undergoing general plane motion can be thought of as rotating about some point called the instantaneous center of rotation. Identifying this point simplifies the analysis since the velocity of any point on the body can be expressed as a product of the angular velocity and its distance from this center.

Link Constraints in Mechanisms

Linkage systems, like a triangular hinged structure, have fixed distances between connected points. These kinematic constraints mean that the motion of one link directly influences the motion of the others. Understanding these constraints is crucial in solving for unknown velocities or displacements in the entire system.

Geometry of Triangles

The properties of triangles, such as side length relationships, angles, and points like midpoints, govern the spatial configuration of the linked system. Geometric relations and trigonometric identities are used to connect the motion of different points in the mechanism, ensuring conformity to the rigid structure.

Vector Decomposition

This technique involves breaking down a vector (such as velocity) into components along chosen directions, often aligned with the geometry of the mechanism (for instance, parallel and perpendicular to a given link). This method is fundamental in analyzing the motion of individual points when they follow different directions, as it allows separate consideration of each component's contribution to the overall movement.

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