Name: Power Transmitted by a Wave on a String
Uploaded: 2019-07-19
Duration: 7 min 4 s
Description: We know that a wave transfers energy from one point to another, but to find the power transmitted by a wave on a string, watch this video.

Question

A hinged construction consists of three rhombs with the ratio of sides $3 : 2 : 1$ .

Vertex $A_{3}$ moves in the horizontal direction at a velocity $v$ . Determine the velocities of vertices $A_{1}, A_{2}$ , and $B_{2}$ at the instant when the angles of the construction are $90 °$ .

Accepted Answer

By hypothesis, the following proportion is preserved between the lengths $l_{1}, l_{2}$ , and $A_{3}$ of segments $v$ , and $A_{0} A_{3}$ during the motion:

$\Rightarrow l_{1} : l_{2} : l_{3} = 3 : (3 + 2) : (3 + 2 + 1) \Rightarrow l_{1} : l_{2} : l_{3} = 3 : 5 : 6$ .

Therefore, the velocities of points $A_{1}, A_{2}$ , and $A_{3}$ are to one another as,

$v_{A_{2}} : v_{A_{1}} : v_{A_{3}} = 3 : 5 : 6$ ,

Hence,

$v_{A_{1}} = \frac{v}{2}, v_{A_{3}} = \frac{5 v}{6}$ .

Let us now consider the velocity of the middle link $(A_{1} B_{2} A_{2} C_{2})$ at the instant when the angles,

of the construction are equal to $90 °$ . In the reference frame moving at a velocity $v_{A_{1}}$ , the velocity $v_{B_{2}}^{'}$ of point $B_{2}$ is directed at this instant along $B_{2} A_{2}$ . The velocity of point $A_{2}$ is directed along the horizontal and is given by

$v_{A_{2}}^{'} = v_{A_{2}} - v_{A_{1}} = \frac{v}{3}$

From the condition of inextensibility of the rod $B_{2} A_{2}$ , it follows that,

$v_{B_{2}}^{'} = v_{A_{2}}^{'} \sin \frac{π}{4} = \frac{v \sqrt{2}}{6}$ .

We can find the velocity of point $B_{z}$ relative to a stationary reference frame by using the cosine law:

$v_{B_{2}}^{2} = v_{A_{1}}^{2} + v_{B_{2}}^{' 2} + (\frac{2 \sqrt{2}}{2}) v_{A_{1}} v_{B_{2}}^{'} = \frac{17}{36} v^{2}$ ,

$v_{B_{2}} = \frac{\sqrt{17}}{6} v_{0}$

A hinged construction consists of three rhombs with the ratio of sides $3 : 2 : 1$ .

Vertex $A_{3}$ moves in the horizontal direction at a velocity $v$ . Determine the velocities of vertices $A_{1}, A_{2}$ , and $B_{2}$ at the instant when the angles of the construction are $90 °$ .

Important Questions on Mechanics

Learn and Prepare for any exam you want !

A hinged construction consists of three rhombs with the ratio of sides 3:2:1. Vertex A3 moves in the horizontal direction at a velocity v. Determine the velocities of vertices A1,A2, and B2 at the instant when the angles of the construction are 90°.

Important Questions on Mechanics

Learn and Prepare for any exam you want !

A hinged construction consists of three rhombs with the ratio of sides $3 : 2 : 1$ .

Vertex $A_{3}$ moves in the horizontal direction at a velocity $v$ . Determine the velocities of vertices $A_{1}, A_{2}$ , and $B_{2}$ at the instant when the angles of the construction are $90 °$ .