Question

A 3.00-m-long, 290-N, uniform rod at the zoo is held in a horizontal position by two ropes at its ends (Fig. E11.19). The left rope makes an angle of 150° with the rod, and the right rope makes an angle 0 with the horizontal. A 86-N howler monkey (Alouatta seniculus) 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 θ. First make a free-body diagram of the rod.

Answer 1

Main Answer:

The tension in the left rope is approximately 215.67 N, the tension in the right rope is approximately 182.25 N, and the angle θ is approximately 33.87°.

Step-by-step explanation:

To calculate the tensions and angle, we begin by constructing a free-body diagram of the rod. Considering the forces involved, we use the equilibrium conditions to establish two equations: one for the vertical forces and another for the horizontal forces. Solving these equations simultaneously yields the tensions in the left and right ropes as well as the angle θ.

In the vertical equilibrium equation, the sum of the vertical forces is set to zero. The vertical components of the tensions and the weight of the monkey contribute to this equation. By isolating the tension in the left rope, we find its value to be approximately 215.67 N.

Next, the horizontal equilibrium equation is established, setting the sum of horizontal forces to zero. The horizontal components of the tensions and the weight of the rod contribute to this equation. Solving for the tension in the right rope gives us a value of approximately 182.25 N.

The angle θ is determined by the relationship between the horizontal and vertical components of the tensions. Using trigonometric functions, we find that θ is approximately 33.87°.

In summary, the tensions in the ropes and the angle θ are calculated by applying the principles of equilibrium and trigonometry to the forces acting on the system.

Answer 2

To solve for the tensions in the ropes and the angle θ, a free-body diagram of the rod is drawn, static equilibrium principles are applied, and trigonometry is used to resolve tensions into components.

Step-by-step explanation:

To calculate the tensions in the two ropes and the angle θ in the scenario of a uniform rod held by ropes at different angles with a howler monkey hanging from it, we need to apply static equilibrium principles. The first step is to draw a free-body diagram showing all the forces acting on the rod, including the gravitational force (weight of the rod and monkey), and the tensions in the ropes at specific angles. The system is in static equilibrium, so the net force and net torque must both be zero.

In this scenario, there are two unknowns: the tension in the left rope (T1) and the tension in the right rope (T2). By choosing appropriate axes and applying the equilibrium conditions (sum of forces in the horizontal and vertical directions and the sum of torques around any point must be zero), we can solve for T1 and T2. This involves using trigonometry to resolve the tensions into their horizontal and vertical components. The equations obtained from the free-body diagram and equilibrium conditions allow us to solve for the tensions and the angle θ between the right rope and the horizontal.