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Chapter 3: Problem 358

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

Expert verified
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

01

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.
02

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: \(T_h = T \cos{\theta}\)
03

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, \(W = T \sin{\theta}\)
04

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) \(\mathrm{T}>\mathrm{W}\).

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