Chapter 5, Problem 5.1DQ

A man sits in a seat that is hanging from a rope. The rope passes over a pulley suspended from the ceiling, and the man holds the other end of the rope in his hands. What is the tension in the rope, and what force does the seat exert on him? Draw a free-body force diagram for the man.

To determine

The tension in the rope and the value of the force exerted by the seat on the man and also draw the free body diagram for the man.

Answer to Problem 5.1DQ

Therefore, the tension in the rope produce is the average of the weight of man and the weight of the seat $\left(T=\frac{W_{\text{man}}+W_{\text{seat}}}{2}\right)$ and the forces exerted by the seat on the man is $\frac{W_{\text{man}}-W_{\text{seat}}}{2}$ and the free body diagram for the man is shown in figure I.

Explanation of Solution

The weight of the man that is sit in a seat and hanging by a rope is act down and produces tension in the rope due to the weight of the man and seat. The free body diagram of the condition is shown below.

Figure I

From the free body diagram of the man plus seat (right side),

$\begin{array}{c}2T=W_{\text{total}}\\ T=\frac{W_{\text{total}}}{2}\end{array}$

• $T$ is the tension produced in the rope.
• $W_{\text{total}}$ is the total weight of the system that is weight of man plus weight of seat.

Substitute $W_{\text{man}}+W_{\text{seat}}$ for $W_{\text{total}}$.

$T=\frac{W_{\text{man}}+W_{\text{seat}}}{2}$

From the free body diagram for the man (left side),

$\begin{array}{c}T+n=W_{\text{man}}\\ n=W_{\text{man}}-T\end{array}$

• $n$ is the normal force exert by the seat on the man.

Substitute $\frac{W_{\text{man}}+W_{\text{seat}}}{2}$ for $T$ to find the value of $n$.

$\begin{array}{c}n=W_{\text{man}}-\frac{W_{\text{man}}+W_{\text{seat}}}{2}\\ =\frac{2W_{\text{man}}-W_{\text{man}}-W_{\text{seat}}}{2}\\ =\frac{W_{\text{man}}-W_{\text{seat}}}{2}\end{array}$

Conclusion:

Therefore, the tension in the rope produce is the average of the weight of man and the weight of the seat $\left(T=\frac{W_{\text{man}}+W_{\text{seat}}}{2}\right)$ and the forces exerted by the seat on the man is $\frac{W_{\text{man}}-W_{\text{seat}}}{2}$ and the free body diagram for the man is shown in figure I.