The resultant of two forces P and Q is at right angles to P, the resultant of P and Q' acting at the same angle is at right angles to Q'. P is

Options

A. A.M.
B. G.M.
C. H.M.
D. None of Q and Q'

Answer

To solve the problem, we need to analyze the forces P, Q, and Q' and their resultant forces. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Given Conditions
We have two forces, P and Q, such that their resultant \( R \) is at right angles to P. This means that the angle between P and R is 90 degrees.

### Step 2: Use the Pythagorean Theorem
Since the resultant \( R \) is at right angles to P, we can use the Pythagorean theorem to express the relationship between P, Q, and R:
\[ R^2 = P^2 + Q^2 \]

### Step 3: Analyze the Second Condition
Now, we consider the second set of forces: P and Q', where the resultant of these two forces is at right angles to Q'. This means that the angle between Q' and the resultant \( R' \) is also 90 degrees.

### Step 4: Apply the Pythagorean Theorem Again
Similar to the first condition, we can express the relationship for the second condition:
\[ R'^2 = P^2 + Q'^2 \]

### Step 5: Relate the Two Scenarios
Since the resultant of P and Q is at right angles to P, and the resultant of P and Q' is at right angles to Q', we can equate the two expressions derived from the Pythagorean theorem:
\[ P^2 + Q^2 = P^2 + Q'^2 \]

### Step 6: Simplify the Equation
By simplifying the equation, we can eliminate \( P^2 \) from both sides:
\[ Q^2 = Q'^2 \]

### Step 7: Conclude the Relationship
From the equation \( Q^2 = Q'^2 \), we can conclude that:
\[ Q = Q' \quad \text{or} \quad Q = -Q' \]
This means that the magnitudes of Q and Q' are equal.

### Final Conclusion
The problem states that P is a force acting at right angles to the resultant of Q and Q'. Therefore, we can conclude that P is a force that maintains the balance of the system, and its value can be determined based on the magnitudes of Q and Q'.