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 horizontal weightless rod of length, $3 l$ is suspended on two vertical strings. Two loads of mass, $m_{1}$ and $m_{2}$ are in equilibrium at equal distances from each other and from the ends of the strings figure.

Determine the tension, $T$ of the left string at the instant when the right string snaps.

Accepted Answer

At the moment of snapping of the right string, the rod is acted upon by the tension $T$ of the left string and the forces $N_{1}$ and $N_{2}$ of normal pressure of loads of mass, $m_{1}$ and $m_{2}$ . Since the rod is weightless (its mass is zero), the equations of its translatory and rotary motions will have the form,

$- T + N_{1} - N_{2} = 0 \Rightarrow N_{1} l = 2 N_{2} l .$

The second equation (the condition of equality to zero of the sum of all moments of force about point $O$ ) implies that

$N_{1} = 2 N_{2} . . . (1)$

Combining these conditions, we get,

$T = N_{1} - N_{2} = N_{2} . . . (2)$

At the moment of snapping of the right string, the accelerations of loads of mass $m_{1}$ and $m_{2}$ will be vertical (point $O$ is stationary, and the rod is inextensible) and connected through the relation,

$a_{2} = 2 a_{1} . . . (3)$

Let us write the equations of motion for the loads at this instant:

$m_{1} g - N_{1}^{'} = m_{1} a_{1} \Rightarrow m_{2} g + N_{2}^{'} = m_{2} a_{2}$ ,

where $N_{1}^{'}$ and $N_{2}^{'}$ are the normal reactions of the rod on loads of mass $m_{1}$ aud $m_{2}$ . Since $N_{1}^{'} = N_{1}$ and $N_{2}^{'} = N_{2}$ , we have,

$m_{1} g - 2 N_{2} = m_{1} a_{1} \Rightarrow m_{2} g + N_{2} = 2 m_{2} a_{1} .$

Hence, we can find $N_{2}$ , and consequently the tension of the string,

$T = N_{2} = \frac{m_{1} m_{2}}{m_{1} + 4 m_{2}} g .$

Important Questions on Mechanics

Learn and Prepare for any exam you want !

A horizontal weightless rod of length, 3l is suspended on two vertical strings. Two loads of mass, m1 and m2 are in equilibrium at equal distances from each other and from the ends of the strings figure. Determine the tension, T of the left string at the instant when the right string snaps.

Important Questions on Mechanics

Learn and Prepare for any exam you want !