What is the angle between two forces of equal magnitude P, if the magnitude of their resultant is `(P)/(2)` ?

Options

A. `cos^(-1)""((5)/(8))`
B. `cos^(-1)""(-(7)/(8))`
C. `cos^(-1)""(-(3)/(8))`
D. `cos^(-1)""((3)/(5))`

Explanation

To find the angle between two forces of equal magnitude \( P \) when the magnitude of their resultant is \( \frac{P}{2} \), we can use the formula for the resultant of two vectors. Here’s the step-by-step solution:

### Step 1: Understand the Problem
We have two forces \( \vec{P} \) and \( \vec{Q} \) with equal magnitudes \( P \). The resultant \( \vec{R} \) of these two forces is given as \( \frac{P}{2} \). We need to find the angle \( \theta \) between the two forces.

### Step 2: Write the Formula for Resultant
The magnitude of the resultant \( R \) of two vectors can be calculated using the formula:
\[ R^2 = P^2 + Q^2 + 2PQ \cos \theta \]
Since \( P = Q \), we can substitute \( Q \) with \( P \):
\[ R^2 = P^2 + P^2 + 2P \cdot P \cos \theta \]
This simplifies to:
\[ R^2 = 2P^2 + 2P^2 \cos \theta \]

### Step 3: Substitute the Known Values
We know that \( R = \frac{P}{2} \). Therefore, we can substitute \( R \) into the equation:
\[ \left(\frac{P}{2}\right)^2 = 2P^2 + 2P^2 \cos \theta \]
Calculating \( \left(\frac{P}{2}\right)^2 \):
\[ \frac{P^2}{4} = 2P^2 + 2P^2 \cos \theta \]

### Step 4: Rearrange the Equation
Now, we can rearrange the equation:
\[ \frac{P^2}{4} = 2P^2(1 + \cos \theta) \]
Dividing both sides by \( P^2 \) (assuming \( P \neq 0 \)):
\[ \frac{1}{4} = 2(1 + \cos \theta) \]

### Step 5: Solve for \( \cos \theta \)
Now, we simplify:
\[ \frac{1}{4} = 2 + 2 \cos \theta \]
Subtracting 2 from both sides:
\[ \frac{1}{4} - 2 = 2 \cos \theta \]
Converting 2 to a fraction:
\[ \frac{1}{4} - \frac{8}{4} = 2 \cos \theta \]
This gives us:
\[ -\frac{7}{4} = 2 \cos \theta \]
Dividing both sides by 2:
\[ \cos \theta = -\frac{7}{8} \]

### Step 6: Find the Angle \( \theta \)
To find \( \theta \), we take the inverse cosine:
\[ \theta = \cos^{-1}\left(-\frac{7}{8}\right) \]

### Final Answer
Thus, the angle between the two forces is:
\[ \theta = \cos^{-1}\left(-\frac{7}{8}\right) \]