Chapter 8, Problem 100A

(a)

To determine

The torque caused by each force.

Answer to Problem 100A

The torque caused by the pole is $5.9\text{ N}\cdot \text{m}$ .

The torque caused by the lamp is $21.1\text{ N}\cdot \text{m}$ .

Explanation of Solution

Given:

The weight of the pole is $27\text{ N}$ .

The weight of the lamp is $64\text{ N}$ .

Formula used:

The expression for the torque is,

  $T=Fr\sin \theta$

Here, $F$ is the applied force, $r$ is the radial distance, and $\theta$ is the angle of the applied force.

Calculation:

The torque caused by the pole is,

  $\begin{array}{l}{\tau }_{\text{Pole}}=F_{\text{Pole}}{r}_{\text{Pole}}\sin \theta \\ {\tau }_{\text{Pole}}=\left(27\text{ N}\right)\times \left(\frac{l_{\text{Pole}}}{2}\right)\sin 90°\\ {\tau }_{\text{Pole}}=\left(27\text{ N}\right)\times \left(\frac{0.44\text{ m}}{2}\right)\sin 90°\\ {\tau }_{\text{Pole}}=5.9\text{ N}\cdot \text{m}\end{array}$

The torque caused by the lamp is,

  $\begin{array}{l}{\tau }_{\text{lamp}}=F_{\text{lamp}}{r}_{\text{lamp}}\sin \theta \\ {\tau }_{\text{lamp}}\left(\text{64 N}\right)\times \left(0.33\text{ m}\right)\sin 90°\\ {\tau }_{\text{lamp}}=21.1\text{ N}\cdot \text{m}\end{array}$

Conclusion:

Therefore, the torque caused by the pole is ${\tau }_{\text{Pole}}=5.9\text{ N}\cdot \text{m}$ and the torque caused by the lamp is ${\tau }_{\text{lamp}}=21.1\text{ N}\cdot \text{m}$ .

(b)

To determine

The tension in the rope supporting the lamp pole.

Answer to Problem 100A

The tension in the rope supporting the lamp pole is $F_{\text{T}}=64\text{ N}$ .

Explanation of Solution

Given:

The weight of the pole is $27\text{ N}$ .

The weight of the lamp is $64\text{ N}$ .

Formula used:

The expression for the torque as follows:

  $T=Fr$

Here, $F$ is the applied force and $r$ is the radial distance.

Calculation:

Show the pivoted lamp pole as shown in figure below.

  

Consider the location of the axis of rotation to make that torque zero.

Consider the vertical component of the tension force as $F_{\text{Ty}}$ .

Apply the total torque on the system.

  $\begin{array}{l}{\tau }_{\text{Cable}}={\tau }_{\text{Pole}}+{\tau }_{\text{lamp}}\\ F_{\text{Ty}}{r}_{\text{Cable}}=F_{\text{Pole}}{r}_{\text{Pole}}+F_{\text{lamp}}{r}_{\text{lamp}}\\ F_T\sin 105°\times 0.44\text{ m}=\left(\text{27 N}\times 0.22\text{ m}\right)+\left(\text{64 N}\times 0.33\text{ m}\right)\\ F_{\text{T}}=64\text{ N}\end{array}$

Conclusion:

Therefore, the tension in the rope supporting the lamp pole is $F_{\text{T}}=64\text{ N}$ .