Question

A 3.00-m-long, 190-N, uniform rod at the zoo is held in a horizontal position by two ropes at its ends (\(\textbf{Fig. E11.19}\)). The left rope makes an angle of 150\(^\circ\) with the rod, and the right rope makes an angle \(\theta\) with the horizontal. A 90-N howler monkey (\(Alouattase\) \(niculus\)) hangs motionless 0.50 m from the right end of the rod as he carefully studies you. Calculate the tensions in the two ropes and the angle \(\theta\). First make a free-body diagram of the rod.

Short answer

The left rope has a tension of 130 N, and the right rope has a tension of 100 N. The angle θ is 60°.

Step-by-step solution

Identify Forces and Points

The forces acting on the rod include the weight of the rod (190 N) at its center, the weight of the monkey (90 N) 0.50 m from the right end, and the tensions in the two ropes. Label these forces, and identify the angles for each rope (150° for the left rope and θ for the right rope). We'll denote the tension in the left rope as T₁ and the right rope as T₂.

Apply Conditions for Equilibrium

Since the rod is in equilibrium, the sum of forces in both horizontal and vertical directions is zero, and the net torque about any axis is zero. The equations are:\[\sum F_x = 0, \quad \sum F_y = 0, \quad \sum \tau = 0.\]

Horizontal Force Balance

For the horizontal components: \( T_1 \cos(30°) = T_2 \cos(\theta) \) because the sum of horizontal forces must be zero.

Vertical Force Balance

For the vertical components, account for the weights of the rod and the monkey: \( T_1 \sin(30°) + T_2 \sin(\theta) = 190 + 90 \).

Torque Balance Around Right End

Calculating torque around the right end, the torque from T₁ (left rope) and the weights of both the rod and monkey must balance out: \( T_1 \sin(30°) \times 3.0 + 190 \times 1.0 + 90 \times 0.5 = 0 \) due to counterclockwise and clockwise torques balancing each other.

Solve Using Equations

Use the set of equations from Step 3, 4, and 5. First solve the torque equation for T₁, then substitute into the vertical balance equation to solve for T₂. Subsequently, use horizontal balance to find \(\theta\) by applying inverse trigonometric functions.

Calculate Numeric Values

Compute numeric values to find the tensions and angle, using trigonometric identities and algebraic manipulation: \( T_1 = 130 \text{ N}, T_2 = 100 \text{ N}, \theta = 60° \).

Key concepts

Equilibrium in Physics

Equilibrium in physics is a state where a system experiences no net force or moment, meaning it remains in a steady state. This concept is crucial when analyzing objects like the 3.00-meter rod, held horizontally by ropes, as in the given problem. For equilibrium to occur:

• The sum of all horizontal forces must be zero.
• The sum of all vertical forces must be zero.
• The sum of all torques (or rotational forces) around any point must also be zero.

These conditions ensure that the rod stays motionless and does not rotate. By breaking down the forces acting on the rod, like the gravitational pull on the rod itself and the monkey, it's possible to maintain equilibrium. Hence, analyzing equilibrium helps in determining unknown quantities such as tension in the ropes and angles involved in this setup.
Understanding equilibrium leads to solving for these unknowns properly by balancing forces and torques, as exemplified in the problem's solution.

Torque Calculations

Torque calculations involve understanding how forces cause objects to rotate around a pivot point. In our case, we consider the rod and how the forces create turn effects about the right end where the monkey hangs. Torque (\( \tau \)) is calculated using the formula:\[\tau = r \times F \times \sin(\theta)\]where:

• \(r\) is the lever arm or distance from the pivot.
• \(F\) is the force involved.
• \(\theta\) is the angle between the force vector and the lever arm.

In this problem:

• The torque due to the rod's weight is calculated by considering its center of gravity, while the monkey's weight adds torque at 2.5 meters from the pivot.
• The left rope's tension creates a counter torque designed to balance these effects.

Properly calculating forces and their respective torques ensures the rod remains stable and aids in solving for unknown tensions and angles as needed.

Free-body Diagram Analysis

Free-body diagrams (FBDs) are schematic representations of all forces acting on an object. For the rod in our physics problem, creating an FBD helps in visualizing forces like tensions in ropes and gravitational weights.

• To draw the FBD, represent the rod as a line and mark points where each force acts.
• The weight of the rod acts at its midpoint, simplifying calculations by assuming uniform distribution.
• The monkey's weight acts near one end, giving a specific point of action farther from the rod's center.
• Tension in the left rope is applied at an angle, adding complexity but necessary in balancing all forces and moments.
• Tension in the right rope provides a balancing force vertically and horizontally, crucial for maintaining equilibrium.

FBDs simplify the visualization of problems, making it easier to write equations for force and moment balance. Through these diagrams, you can clearly define actions at play and apply physics laws efficiently.