Question

A crow perches on a clothesline midway between two poles. Each end of the rope makes an angle of \(\theta\) below the horizontal where it connects to the pole. If the weight of the crow is \(W,\) what is the tension in the rope? Ignore the weight of the rope.

Short answer

Answer: \(T = \frac{W}{2 \sin(\theta)}\)

Step-by-step solution

Draw a free body diagram of the situation

To better visualize the forces acting on the situation, let's draw a free body diagram of the rope, crow, and tension forces from each pole. Two tension forces, T1 and T2, act at each end of the rope making an angle of \(\theta\) below the horizontal, and the weight of the crow W acts downwards in the middle.

Write down the vertical and horizontal force equilibrium equations

Using equilibrium conditions, we can write the vertical and horizontal force equations: Vertical forces: \(T1 \sin(\theta) + T2 \sin(\theta) - W = 0\) (1) Horizontal forces: \(T1 \cos(\theta) - T2 \cos(\theta) = 0\) (2)

Solve the horizontal force equation for T1 or T2

Since \(T1 \cos(\theta) = T2 \cos(\theta)\), we can divide both sides of the equation by \(\cos(\theta)\) to find: \(T1 = T2\)

Substitute T1 into the vertical force equation

Now we can substitute \(T2\) for \(T1\) in the vertical force equation (1) to get: \(2T2 \sin(\theta) - W = 0\)

Solve for T2 (or T1)

Rearranging the equation and solving for \(T2\), we get: \(T2 = \frac{W}{2 \sin(\theta)}\)

Calculate the tension T in the rope

Since \(T1 = T2\), we can conclude that the tension T in the rope is: \(T = \frac{W}{2 \sin(\theta)}\) Now, we have found the tension T in the rope as a function of the weight of the crow W and the angle \(\theta\).