Problem 17 - College Physics

Question

Telephone and electrical lines are allowed to sag between poles so that the tension will not be too great when something hits or sits on the line. (a) Is it possible to have the lines perfectly horizontal? Why or why not? (b) Suppose that a line were stretched almost perfectly horizontally between two poles that are \(30 \mathrm{~m}\) apart. If a \(0.25-\mathrm{kg}\) bird perches on the wire midway between the poles and the wire sags \(1.0 \mathrm{~cm},\) what would be the tension in the wire? (Neglect the mass of the wire.)

Step-by-Step Solution

Answer

(a) No, perfectly horizontal lines require infinite tension. (b) The tension is approximately 3675 N.

Step 1: Understanding Tension and Angle

For perfectly horizontal lines, the angle of sag would be zero. If the line cannot be horizontal, it suggests that no finite tension could support any load at zero angle, as it would require an infinite tension force to counteract any additional weight or force applied onto the line.

Step 2: Analyzing Forces on the Wire (Part B)

We assume the center of the wire sags by 1 cm due to the weight of a bird. This creates a right angle triangle with the sag and half the distance between poles (30/2 = 15 m) as legs. Applying trigonometry, the horizontal tension forces balance the bird's downward force.

Step 3: Calculating Sag Component

We use the vertical component of the tension (T_y) and equate it to the bird's weight. If T is the tension in the wire, then T_y can be expressed by trigonometric relations as T sin(θ). The angle θ can be calculated using sag and half the distance (0.01 m/15 m = tan(θ)).

Step 4: Calculating Bird's Weight

The bird's weight (W) is calculated using W = mg, where m = 0.25 kg and g = 9.81 m/s². Therefore, W = 0.25 kg * 9.81 m/s² = 2.4525 N.

Step 5: Using Trigonometric Relations

With tan(θ) = opposite/adjacent = 0.01 m / 15 m, we find sin(θ) ≈ tan(θ) ≈ 0.01/15. We estimate sin(θ) = 0.000667 since the sag is small.

Step 6: Solving for Tension in the Wire

Equating T_y to the bird's weight: T · 0.000667 = 2.4525 N. Solving for T, we get T ≈ 2.4525 / 0.000667 ≈ 3675 N.

Key Concepts

Sag tension problem

In the context of physics, when we talk about tension in wires or ropes, it's crucial to consider the concept of sag tension. This tension is particularly important in situations where wires or ropes are used to span distances, such as between telephone poles or electricity towers. Sag happens because gravity pulls down on the line, creating a natural curve. Not allowing a line to sag would mean it needs to be perfectly straight, requiring infinite tension to support any weight on it. This is why you never see perfectly horizontal wires; they couldn't physically handle the added weight without breaking.

Trigonometrical analysis in physics

Trigonometry is a powerful tool in physics used to analyze forces in cases like the sag tension problem. When a bird perches on a wire, the sag creates a right triangle where:

• One leg is the sag amount (in this case, 1 cm or 0.01 m),
• The other is half the distance between the poles, 15 m.

By understanding this right-angle triangle, we can use trigonometric ratios like tangent to find angles or components of force. In this exercise, tan(θ) is used as the ratio of the sag (the opposite side) over half the wire's span (the adjacent side), providing a way to calculate angles and tensions accurately.

Physics problem-solving steps

Solving physics problems involves a systematic approach to understanding and calculating relevant quantities. In this exercise, we followed several key steps:

• First, comprehend the limitation of perfectly horizontal lines and why some sag is necessary.
• Next, relate the forces acting on the system, like how the bird contributes to the sag by providing a downward force.
• Then, break down these forces using trigonometry to find the relationship between the wire's tension and the bird's weight.
• Finally, calculate the necessary quantities, like the angle and the tension, to fully understand the physics of the problem.

This structured method helps to tackle even the most complex scenarios you’ll encounter in physics, ensuring nothing is overlooked.

Bird's weight calculation

To understand the effect a bird has on a wire, you must first calculate its weight. Weight is determined using the formula: \[ W = mg \]where \( m \) is the mass (0.25 kg for our bird) and \( g \) is the acceleration due to gravity (9.81 m/s² on Earth). Put these together, and you calculate the bird's weight:\[ W = 0.25 \times 9.81 = 2.4525 \text{ N} \]This number helps us quantify the force the bird applies downwards on the wire, a necessary step to figure out how much tension the wire will experience. Knowing the bird's weight allows you to equate it to the vertical component of tension (\( T_y \)) and use trigonometry to solve for the total tension in the wire, as we did in this exercise.