A hinged construction consits of three rhombus with the ratio of sides (5:3:2) . Vertex A3 moves in the horizontal direction with velocity V . Velocity of A2 will be :- A) 2.5 V B) 1.5V. C) 2/3 V. D) 0.8 V

To determine the velocity of point A2 in relation to the velocity of point A3, we need to analyze the geometry and motion of the hinged construction made up of three rhombuses. Given that the sides of the rhombuses are in the ratio of 5:3:2, we can denote the lengths of the sides as 5x, 3x, and 2x, respectively. Let's break down the problem step by step.

Understanding the Geometry

In a hinged construction, the movement of one point affects the others due to the constraints of the hinges and the lengths of the sides. Here, we have three rhombuses connected at their vertices. The vertices are labeled A1, A2, and A3, with A3 moving horizontally with a velocity V.

Analyzing the Motion

Since A3 is moving horizontally, we can visualize how this motion translates through the rhombuses to A2. The key is to understand that the angles within the rhombuses remain constant, and the sides maintain their lengths. This means that as A3 moves, A2 will also move, but the velocity will depend on the ratios of the sides.

Using Ratios to Find Velocity

Let’s denote the lengths of the sides of the rhombuses as follows:

• Side connecting A2 to A1: 5x
• Side connecting A1 to A3: 3x
• Side connecting A2 to A3: 2x

Since A3 moves horizontally, we can use the concept of similar triangles or the properties of rhombuses to relate the velocities. The velocity of A2 can be derived from the ratio of the sides connected to A2 and A3. The relevant sides are 2x (A2 to A3) and 3x (A1 to A3).

Calculating the Velocity of A2

Using the ratios of the sides, we can set up the relationship:

Velocity of A2 (V2) = (Length of side A2 to A3 / Length of side A1 to A3) * Velocity of A3

Substituting the lengths:

V2 = (2x / 3x) * V = (2/3) * V

Final Result

Thus, the velocity of A2 in relation to the velocity of A3 is:

V2 = (2/3)V

This means the correct answer is C) 2/3 V.

In summary, by analyzing the geometric relationships and applying the ratios of the sides of the rhombuses, we can effectively determine the velocity of point A2 based on the known velocity of point A3. This method illustrates how interconnected movements in mechanical systems can be analyzed using basic principles of geometry and ratios.