91Ó°ÊÓ

A tension force is a force created when a rope, string, or cable is stretched as a result of an external force. Along the length of the rope or cable, tension is exerted in the opposite direction as the force acting on it.

Here we have given the distance from the top to the cord is $\mathrm{d}=40.0\text{″¾}$

The length of the pole is $\mathrm{L}=1.75\text{″¾}$

The weight of the pole is ${\mathrm{w}}_{\mathrm{p}}=55.0\text{ N}$

The maximum outward force on the nail is ${\mathrm{F}}_{\max }=22.0\text{ N}$

The angle between the wire and wall is$\mathrm{θ}=37°$

(a)

Consider the following figure,

Here, the force exerted by the hinge, in general, has both horizontal and vertical components.

Let${\mathrm{F}}_{\mathrm{h},\mathrm{y}}$and ${\mathrm{F}}_{\mathrm{h},\mathrm{x}}$ be the vertical and horizontal force exerted by the hinge respectively.

We know that the horizontal component of the tension force equals to ${\mathrm{F}}_{\max }$.

So we have,

Now, substitute the values in the above equation. We get,

Now, the net torque about the hinge point has to be zero.

So, $(\mathrm{L}−\mathrm{d})â‹\dots \mathrm{W}−\mathrm{L}â‹\dots \mathrm{°Õ²õ¾\pm ²Ôθ}=0$

Hence, the greatest weight $W$that can be supported this way without pulling out the nail is $28.5\text{ N}$.

(b)

Now, we want to find the magnitude of the force exerted by the hinge,${\mathrm{F}}_{\mathrm{h}}$.

By, the first condition of equilibrium, the net force must be zero both horizontally and vertically.

So we have,

$\begin{array}{c}−{\mathrm{w}}_{\mathrm{p}}−\mathrm{°Õ³¦´Ç²õθ}+{\mathrm{F}}_{\mathrm{h},\mathrm{y}}=0\\ {\mathrm{F}}_{\mathrm{h},\mathrm{y}}=55.0\text{ N}+\left(36.6\text{ N}\right)\cos 37.0°\\ {\mathrm{F}}_{\mathrm{h},\mathrm{y}}=84.2\text{ N}\end{array}$

The magnitude of the hinge force is given by

$\begin{array}{c}{\mathrm{F}}_{\mathrm{h}}=\sqrt{{{\mathrm{F}}_{\mathrm{h},\mathrm{x}}}^2+{{\mathrm{F}}_{\mathrm{h},\mathrm{y}}}^2}\\ =\sqrt{{(−6.50\text{ N})}^2+{\left(84.2\text{ N}\right)}^2}\\ =84.5\text{ N}\end{array}$

The magnitude of the force that the hinge exerts on the pole $84.5\text{ N}$.