A hinged construction consists of three rhomb | vedclass

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(D) $1)$ We know that the diagonal of a rhombus is proportional to its side length.$\Rightarrow$ The corresponding ratio of the diagonals is $5:3:2$.L

Vedclass · Accepted Answer

$0.8$

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(D) $1)$ We know that the diagonal of a rhombus is proportional to its side length.$\Rightarrow$ The corresponding ratio of the diagonals is $5:3:2$.Let the lengths of the diagonals be $5y, 3y,$ and $2y$ respectively,as shown in the figure.$2)$ Now,the velocity of $A_3$ relative to $A_0$ is given by the rate of change of the total length $A_0 A_3$:$V_{A_3} - V_{A_0} = \frac{d(A_0 A_3)}{dt}$Since $A_0$ is fixed,$V_{A_0} = 0$. The total length $A_0 A_3 = 5y + 3y + 2y = 10y$.$V - 0 = \frac{d(10y)}{dt}$$V = 10 \frac{dy}{dt} \dots (1)$$3)$ Similarly,the velocity of $A_3$ relative to $A_2$ is the rate of change of the length $A_2 A_3$:$V_{A_3} - V_{A_2} = \frac{d(A_2 A_3)}{dt}$$V - V_{A_2} = \frac{d(2y)}{dt}$$V - V_{A_2} = 2 \frac{dy}{dt} \dots (2)$$4)$ From equation $(1)$,we have $\frac{dy}{dt} = \frac{V}{10}$.Substituting this into equation $(2)$:$V - V_{A_2} = 2 \left( \frac{V}{10} \right)$$V - V_{A_2} = 0.2V$$V_{A_2} = V - 0.2V = 0.8V$.Thus,the velocity of $A_2$ is $0.8V$.

$A$ hinged construction consists of three rhombs with the side ratio $5:3:2$. Vertex $A_3$ moves in the horizontal direction at a velocity $v$. The velocity of $A_2$ is ....... $V$.

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