Question

A sparrow flying in air sits on a stretched telegraph wire. If the weight of the sparrow is \(\mathrm{W}\), which of the following is true about the tension T produced in the wire? (A) \(\mathrm{T}=\mathrm{W}\) (B) \(\mathrm{T}<\mathrm{W}\) (C) \(\mathrm{T}=0\) (D) \(\mathrm{T}>\mathrm{W}\)

Short answer

The tension in the wire (T) must be greater than the weight of the sparrow (W). This is because the tension has both vertical and horizontal components, and the vertical component needs to balance the sparrow's weight. The relationship between the tension and weight is given by \(T = \frac{W}{\sin{\theta}}\), where θ is the angle between the wire and the horizontal direction. So, the correct option is (D) \(\mathrm{T}>\mathrm{W}\).

Step-by-step solution

Identify the forces acting on the wire

When the sparrow sits on the wire, two forces come into play: 1. Weight of the sparrow (W) - This acts vertically downwards. 2. Tension in the wire (T) - This force is produced within the wire as a reaction to the weight of the sparrow, and acts along the direction of the wire.

Analyze the vertical and horizontal components of the tension

Since the wire is stretched, it must be straight. Let's assume the wire is inclined at an angle θ with the horizontal direction. We can decompose the tension force T into its vertical and horizontal components: - Vertical component of tension: \(T_v = T \sin{\theta}\) - Horizontal component of tension: \(T_h = T \cos{\theta}\)

Determine the relationship between the weight of the sparrow and the vertical component of tension

Since the sparrow is in equilibrium while sitting on the wire, the vertical component of the tension must balance the weight of the sparrow: \(T_v = W\) As we know from step 2, the vertical component of tension is given by \(T \sin{\theta}\). Hence, \(W = T \sin{\theta}\)

Compare the tension in the wire with the weight of the sparrow

We can rewrite the relationship obtained in step 3 as: \(T = \frac{W}{\sin{\theta}}\) Since θ is a nonzero angle (the wire is not perfectly horizontal), this means that \(0 < \sin{\theta} \leq 1\). Therefore, the tension in the wire must be greater than the weight of the sparrow: \(T > W\) Hence, the correct option is (D) \(\mathrm{T}>\mathrm{W}\).

Key concepts

Mechanics

Mechanics is the branch of physics that deals with the motion and the forces acting on objects. It helps us understand how objects move and interact in our world. In the context of the exercise, mechanics is used to describe the situation of a sparrow sitting on a telegraph wire. The sparrow's weight exerts a force on the wire, which is part of the forces mechanics studies. Understanding how these forces work helps us to analyze what happens when different objects and bodies exert forces on each other, like the sparrow and the wire.

Forces

Forces are pushes or pulls that act on an object. When we talk about forces, we often refer to them in terms of their magnitude (how strong they are) and direction (where they are pointing). In the exercise, there are two main forces: the weight of the sparrow, which acts vertically downwards, and the tension in the wire, which acts along the wire.
- **Weight of the Sparrow (W):** This force is due to gravity pulling the sparrow downwards towards the Earth. This is a vertical force. - **Tension in the Wire (T):** This is a force that arises within the wire as a reaction to the weight of the sparrow. Tension is important as it helps keep objects like the wire stable. These forces are key to solving physics problems involving objects suspended in air or hanging across other objects.

Equilibrium

Equilibrium occurs when all the forces acting on an object are balanced, giving the object a constant velocity—typically, it remains at rest. For the sparrow sitting on the wire, equilibrium ensures that the wire does not collapse under its weight. When a sparrow perches, its weight is balanced by the tension in the wire's vertical component. This is known as static equilibrium because the sparrow and wire aren't moving. For equilibrium in this instance, the key condition is that the vertical components of forces must balance:

• Vertical component of tension, which is equal to the weight (\(T_v = W\)).
• This leads to the realization: \(T imes \sin(\theta) = W\).

Understanding this balance helps in predicting how structures will respond under various loads.

Decomposition of Forces

When forces are not aligned with the axis of interest, they are decomposed into components along the axes. This simplifies the analysis of the forces acting on the system. In mechanics, we often break down a force into perpendicular components, called the horizontal and vertical components, using trigonometric functions.
For the sparrow and wire:

• **Vertical Component (\(T_v\)):** This is \(T \sin(\theta)\) and it balances with the weight of the sparrow.
• **Horizontal Component (\(T_h\)):** This is \(T \cos(\theta)\) which is not balanced by another force in this scenario but provides stability to the wire.

By examining the force components, we can more easily understand how and why the tension in the wire exceeds the weight of the sparrow, as demonstrated by leveraging the relationship \(T = \frac{W}{\sin(\theta)}\). This decomposition is crucial in designing and analyzing structures and systems in physics and engineering.