Question

The setup shown consists of two homogeneous rods each of mass of mass $m = 72$ kg, hinged on fixed supports. Two men each of mass $3m$ standing on the lower rod keep the setup in equilibrium by applying appropriate pulls on the ropes. Assume the ropes and the pullies to be ideal and neglect any friction at the axles of the hinges.

Answer 1

The question is incomplete, as it does not ask for a specific quantity or condition for equilibrium.

Answer 2

The problem statement describes a system in equilibrium but does not pose a specific question. It provides the masses of the rods and men, and states that the ropes and pulleys are ideal, and friction at hinges is negligible. Without a specific question (e.g., "What is the tension in the ropes?", "What are the hinge reactions?", "What is the minimum pull required?"), a definitive numerical answer cannot be provided.

However, based on the context of such problems, it's highly probable that the question implicitly asks for the tension (pull) in the ropes required to maintain equilibrium. Let's analyze the forces involved:

Let:

• $m_r = m = 72$ kg be the mass of each rod.
• $m_m = 3m = 3 \times 72 = 216$ kg be the mass of each man.
• $T$ be the tension in each rope.
• $g$ be the acceleration due to gravity.

1. Free Body Diagram of a Man: Each man is standing on the lower rod and pulling a rope.

• Downward force: Weight of the man, $W_m = m_m g = 3mg$.
• Upward forces: Normal force from the lower rod, $N$, and tension from the rope, $T$. For equilibrium of a man: $N + T = 3mg$ $N = 3mg - T$

2. Free Body Diagram of the Lower Rod: The lower rod has mass $m_r = m$. It is hinged to fixed supports. Two men stand on it.

• Downward force: Weight of the rod, $W_r = mg$.
• Downward forces from the men: $2N$ (since there are two men).
• Upward forces: Vertical components of hinge reactions, let's call their sum $V_L$.
• The diagram shows the other end of the rope, after passing over the pulley on the upper rod, is attached to the lower rod. So, each rope exerts an upward tension $T$ on the lower rod. There are two such ropes. For vertical equilibrium of the lower rod: $V_L + 2T = mg + 2N$ Substitute $N = 3mg - T$: $V_L + 2T = mg + 2(3mg - T)$ $V_L + 2T = mg + 6mg - 2T$ $V_L = 7mg - 4T$

3. Free Body Diagram of the Upper Rod: The upper rod has mass $m_r = m$. It is hinged to fixed supports. Two pulleys are attached to it.

• Downward force: Weight of the rod, $W_r = mg$.
• Downward forces from the pulleys: Each pulley has a rope passing over it. One end of the rope goes to the man, the other to the lower rod. The tension in both segments of the rope is $T$. So, each pulley exerts a downward force of $2T$ on the upper rod. Since there are two pulleys, the total downward force from pulleys is $2 \times 2T = 4T$.
• Upward forces: Vertical components of hinge reactions, let's call their sum $V_U$. For vertical equilibrium of the upper rod: $V_U = mg + 4T$

4. Free Body Diagram of the Entire System (Two Rods + Two Men): The total mass of the system is $m_{total} = m_r + m_r + m_m + m_m = m + m + 3m + 3m = 8m$.

• Downward force: Total weight $= 8mg$.
• Upward forces: Vertical hinge reactions from the fixed supports for the upper rod ($V_U$) and the lower rod ($V_L$). For vertical equilibrium of the entire system: $V_U + V_L = 8mg$

Now, substitute the expressions for $V_U$ and $V_L$: $(mg + 4T) + (7mg - 4T) = 8mg$ $8mg = 8mg$

This equation is an identity. This means that the system is in equilibrium for any value of tension $T$, as long as the normal force $N$ is non-negative ($N \ge 0$) and the tension $T$ is non-negative ($T \ge 0$). From $N = 3mg - T$, for $N \ge 0$, we must have $3mg - T \ge 0$, which means $T \le 3mg$. So, the tension $T$ can be any value in the range $0 \le T \le 3mg$.

Since the question asks for "appropriate pulls on the ropes" and does not specify any additional conditions (like minimum pull, zero reaction at a hinge, specific height, etc.), there isn't a unique numerical answer for $T$. The problem statement is incomplete in this regard.

However, in many such problems, if a unique value is expected, it usually implies a condition where the men are "just barely" standing, i.e., $N=0$, or the system is symmetrical and balanced in a way that leads to a unique solution. If $N=0$, then $T = 3mg$. In this case, the men are lifting themselves entirely by the ropes, and are not pressing on the lower rod. If $T = 3mg$: $N = 0$ $V_L = 7mg - 4(3mg) = 7mg - 12mg = -5mg$. A negative vertical reaction means the hinge is pulling down, which is possible. $V_U = mg + 4(3mg) = mg + 12mg = 13mg$.

Without a specific question or additional constraint, we cannot determine a unique value for $T$. If the question implies that the men are just supporting themselves (i.e., normal force from the rod is zero), then $T=3mg$. Given $m = 72$ kg, $g \approx 9.8 \text{ m/s}^2$: $T = 3 \times 72 \times 9.8 = 216 \times 9.8 = 2116.8$ N.

However, since no such condition is stated, the problem is under-specified. If it were a multiple-choice question, the options might hint at the specific condition implied. As it is, any $T$ such that $0 \le T \le 3mg$ would maintain equilibrium.

The problem asks for "appropriate pulls". This could mean the pulls that allow equilibrium to be maintained. Since any value of $T$ in the range $0 \le T \le 3mg$ is appropriate, the question is ill-posed for a single numerical answer without further constraints.

Let's assume the most common interpretation where one is expected to calculate the tension required. Since the system is in equilibrium for a range of tensions, there might be a missing condition.

Given the common nature of such problems, it's possible that the question is implicitly asking for the force required to keep the system just in equilibrium, or a specific configuration. Without further information, it's impossible to give a unique numerical answer.

The problem is likely from a context where a specific condition (e.g., $N=0$ or a specific hinge reaction) is implied or given in a subsequent part of the question which is not visible here.

The provided information is insufficient to determine a unique value for the pull on the ropes.