Question

Question: An archer’s bow is drawn at its midpoint until the tension in the string is equal to the force exerted by the archer. What is the angle between the two halves of the string?

Short answer

Answer:

The angle between the two halves of the string is 120⁰.

Step-by-step solution

Understanding the given information

The tension in the string = Force exerted by the archer.

Concept and formula used in the given question

Using the free body diagram, you can write the relation between F and T. Then to satisfy the given condition, you can find the value of the angle made by each string segment with the vertical. You can then find the angle between the two halves of the string. The formulas used are given below.

Newton’s second law,

$F_{net}=ma$

At static equilibrium,

$F_{net}=0$

Calculation for the angle between the two halves of the string

Free body diagram:

At equilibrium,

$F=2T\text{Sin}\theta$

Given that

F = T

To obtain this, the condition is

$$\begin{gathered} \text{Sin}\theta =\frac{1}{2} \\ \theta =30° \end{gathered}$$

But, is the angle made by each segment of the string with the vertical. So, the angle between two halves of the string is

$$\begin{gathered} f=180°-2\theta \\ f=180°-60° \\ f=120° \end{gathered}$$

Therefore, the angle between the two halves of the string is 120°.