A given force is resolved into components P and Q equally inclined to it. Then

Options

A. P=2Q
B. 2P=Q
C. P=Q
D. None of these

Answer

To solve the problem of resolving a given force \( F \) into components \( P \) and \( Q \) that are equally inclined to it, we can follow these steps:

### Step 1: Understand the Geometry of the Problem
When a force \( F \) is resolved into two components \( P \) and \( Q \), and both components are equally inclined to the force, it means that the angle between the force and each component is the same. Let's denote this angle as \( \alpha \).

### Step 2: Set Up the Relationship Between the Forces
Since \( P \) and \( Q \) are equally inclined to \( F \), we can express the components in terms of the angle \( \alpha \):
- The horizontal component of \( F \) can be expressed as \( F \cos(\alpha) \).
- The vertical component of \( F \) can be expressed as \( F \sin(\alpha) \).

### Step 3: Equate the Components
Since both components \( P \) and \( Q \) are equal and inclined at the same angle \( \alpha \), we can write:
- \( P = F \cos(\alpha) \)
- \( Q = F \sin(\alpha) \)

### Step 4: Use the Property of Equal Angles
Since \( P \) and \( Q \) are equal, we can set \( P = Q \):
\[ F \cos(\alpha) = F \sin(\alpha) \]

### Step 5: Simplify the Equation
Dividing both sides by \( F \) (assuming \( F \neq 0 \)):
\[ \cos(\alpha) = \sin(\alpha) \]

### Step 6: Solve for the Angle
The equation \( \cos(\alpha) = \sin(\alpha) \) implies that:
\[ \tan(\alpha) = 1 \]
This means:
\[ \alpha = 45^\circ \]

### Step 7: Calculate the Components
Now, substituting \( \alpha = 45^\circ \) back into the equations for \( P \) and \( Q \):
\[ P = F \cos(45^\circ) = F \cdot \frac{1}{\sqrt{2}} = \frac{F}{\sqrt{2}} \]
\[ Q = F \sin(45^\circ) = F \cdot \frac{1}{\sqrt{2}} = \frac{F}{\sqrt{2}} \]

### Conclusion
Thus, the components \( P \) and \( Q \) are both equal to \( \frac{F}{\sqrt{2}} \) when the force \( F \) is resolved into two equal components that are inclined at \( 45^\circ \) to the force.

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