Chapter 2, Problem 16TE

Harry the painter swings year after year from his bosun’s chair. His weight is $500\text{N}$ and the rope, unknown to him, has a breaking point of $300\text{N}$. Why doesn’t the rope break when he is supported as shown at the left below? One day Harry is painting near a flag pole, and for a change he ties the free end of the rope to the flagpole instead of to his chair, as shown at the right. Why did Harry end up taking his vacation early?

To determine

The reason, of the rope not breaking and Harry not taking his vacation early.

Answer to Problem 16TE

The reason that the rope is not breaking is that the tension on rope is $250\text{N}$ which is less than breaking point $300\text{N}$ and the painter is not safe and the rope will snap resulting in injuries to him because the tension on rope is $500\text{N}$ which is more than breaking point $300\text{N}$.

Explanation of Solution

Given Info: The weight of Harry is $500\text{N}$ and the breaking point of rope is $300\text{N}$.

The free body diagram of the system in left diagram is shown below.

The tension in both ends of rope are $T_1$ and $T_2$, whereas the weight of the painter is $W$.

The person hangs in figure one will be in mechanical equilibrium and net force acting on the body is zero. The tension acting on the rope will be equal because person hangs from single rope and the friction is neglected, that is $T_1=T_2=T$.

Write the force balance equation for the system.

$\begin{array}{c}{\displaystyle \sum F}=0\\ 2T+W=0\\ T=\left(-\frac{W}{2}\right)\end{array}$

Here,

$W$ is the weight of the person

$W$ is the tension of the rope

Substitute $-500\text{N}$ for $W$ to find $T$.

$\begin{array}{l}T=\left(-\frac{\left(-500\text{N}\right)}{2}\right)\\ T=250\text{}\text{N}\end{array}$

Thus, the painter is safe because the tension on rope is $250\text{N}$ which is less than breaking point $300\text{N}$.

The free body diagram of the system in right diagram is shown below.

The tension in the rope in this case is $T$'.

The net force on the person will be zero that is the sum of tension and weight of person is zero. The painter is supported by one section of the rope.

Write the force balance equation for the system.

$\begin{array}{c}{T}^{\prime }+W=0\\ T^{\prime }=-W\end{array}$

Substitute $-500\text{N}$ for $W$ to find $T$'.

$\begin{array}{l}T=-\left(-500\text{N}\right)\\ T=500\text{}\text{N}\end{array}$

Thus, the painter is not safe and the rope will snap resulting in injuries to him because the tension on rope is $500\text{N}$ which is more than breaking point $300\text{N}$.

Conclusion:

Therefore, the painter is safe because the tension on rope is $250\text{N}$ which is less than breaking point $300\text{N}$ and the painter is not safe and the rope will snap resulting in injuries to him because the tension on rope is $500\text{N}$ which is more than breaking point $300\text{N}$