Q58P - University Physics with Modern Physics

Question

Two ropes in a vertical plane exert equal-magnitude forces on a hanging weight but pull with an angle of 72.0° between them. What pull does each rope exert if their resultant pull is 372 N directly upward?

Step-by-Step Solution

Answer

The pull exerted by each rope is $230 \mathrm{N}.$

Identification of given data

The given data can be listed below,

• The angle at which ropes are pulled is, $\theta =72°$.
• The resultant pull exerted is, $\mathrm{F}=372 \mathrm{N}$.

Significance of the force

Force is equal to the rate of momentum change with respect to time in any given situation, which facilitates the acceleration of an object

Determination of pull exerted by the ropes

The magnitude of resultant of the components of two vectors in the y-direction is given by,

${\mathrm{C}}_{\mathrm{y}}={\mathrm{A}}_{\mathrm{y}}+{\mathrm{B}}_{\mathrm{y}}$

Here,  $A_y$ is the magnitude of y-component of force $\stackrel{\rightharpoonup }{A}$, $B_y$ is the magnitude of y-component of force $\stackrel{\rightharpoonup }{B}$
and $C_y$ is the magnitude of resultant of forces,$\stackrel{\rightharpoonup }{A}$ and $\stackrel{\rightharpoonup }{B}$.

Assuming y-axis lies along the north, the three vectors, their magnitudes and their direction are shown in the diagram given below,

                             

                                   

The magnitude of the two vectors is equal. and they make equal angle $\theta$ with the y-axis.

Therefore,

$$\begin{gathered} {\mathrm{A}}_{\mathrm{y}}={\mathrm{B}}_{\mathrm{y}} \\ =\mathrm{Acos}\left(\mathrm{\theta }\right) \\ =\mathrm{Acos} {36}^{°} \end{gathered}$$

Substitute values in the equation of resultant force, the value of A is given by,

 $$\begin{gathered} \mathrm{A} \mathrm{cos\theta }+\mathrm{B} \mathrm{cos\theta }={\mathrm{C}}_{\mathrm{y}} \\ \mathrm{A} \cos 36°+\mathrm{A} \cos 36°=372 \\ 2\mathrm{A} \cos 36°=372 \\ \mathrm{A}=230 \mathrm{N} \end{gathered}$$

Thus, the pull exerted by each rope is $230 \mathrm{N}$.