Chapter 2, Problem 23A

Burl and Paul stand on their sign-painting scaffold. Tension in the left rope is measured a scale. Rank the tensions in that rope from greatest to least.

To determine

To rank:The tensions in the ropes from greatest to least.

Answer to Problem 23A

The rank of the tensions in the ropes from greatest to least is $T_{\text{C}}>T_{\text{D}}>T_{\text{A}}=T_{\text{B}}$ .

Explanation of Solution

Given:

The given figure is shown below.

  

Formula used:

The expression for the weight is,

  $W=mg$

Here, $m$ is the mass and $g$ is the acceleration due to gravity.

Calculation:

Take the weight of person B as $W_{\text{B}}$ and person P as $W_{\text{P}}$ .

For case A:

Person B is at the left of the scaffold and person P is at the right of the scaffold.

The sum of the weight of the person B and P is equally distributed between the two ropes.

The tension in the rope is,

  $\begin{array}{l}T=W\\ T_{\text{A}}=\left(\frac{W_{\text{B}}+W_{\text{P}}}{2}\right)\end{array}$

For case B:

Person B and P are at the middle of the scaffold.

The sum of the weight of person B and P is equally distributed between the two ropes.

The tension in the rope is,

  $\begin{array}{l}T=W\\ T_{\text{B}}=\left(\frac{W_{\text{B}}+W_{\text{P}}}{2}\right)\end{array}$

For case C:

Person B and P are at the left of the scaffold.

The weights of person B and P are balanced by the tension in the rope.

The tension in the rope is,

  $\begin{array}{l}T=W\\ T_{\text{C}}=W_{\text{B}}+W_{\text{P}}\end{array}$

For case D:

Person B is at the left of the scaffold and person P is at the middle of the scaffold.

The weight of person B and half the weight of person P are balanced by the tension in the rope.

The tension in the rope is,

  $\begin{array}{l}T=W\\ T_{\text{D}}=W_{\text{B}}+\frac{W_{\text{P}}}{2}\end{array}$

By comparing the values of the tension in the ropes $T_{\text{C}}>T_{\text{D}}>T_{\text{A}}=T_{\text{B}}$ .

Conclusion:

Thus, the rank of the tensions in the ropes from greatest to least is $T_{\text{C}}>T_{\text{D}}>T_{\text{A}}=T_{\text{B}}$ .