Two forces P and Q act at such an angle that R=P. If P is doubled, the new resultant is at right angles to Q.

Answer

To solve the problem step by step, we need to analyze the forces and the conditions given in the question.

### Step 1: Understand the Given Information
We have two forces, P and Q, acting at an angle θ such that the resultant R is equal to P. This means:
\[ R = P \]

### Step 2: Use the Resultant Force Formula
The formula for the resultant of two forces P and Q acting at an angle θ is given by:
\[ R = \sqrt{P^2 + Q^2 + 2PQ \cos \theta} \]
Since we know that \( R = P \), we can set up the equation:
\[ P = \sqrt{P^2 + Q^2 + 2PQ \cos \theta} \]

### Step 3: Square Both Sides
Squaring both sides gives:
\[ P^2 = P^2 + Q^2 + 2PQ \cos \theta \]

### Step 4: Simplify the Equation
Subtract \( P^2 \) from both sides:
\[ 0 = Q^2 + 2PQ \cos \theta \]
This can be rearranged to:
\[ Q^2 = -2PQ \cos \theta \]

### Step 5: Analyze the Situation When P is Doubled
Now, we double the force P, so it becomes \( 2P \). The new resultant \( R' \) is at right angles to Q. When the resultant is at right angles to Q, the angle between R' and Q is 90 degrees, which means:
\[ \cos(90^\circ) = 0 \]

### Step 6: Write the New Resultant Force Equation
Using the resultant formula again for the new forces:
\[ R' = \sqrt{(2P)^2 + Q^2 + 2(2P)Q \cos(90^\circ)} \]
Since \( \cos(90^\circ) = 0 \), this simplifies to:
\[ R' = \sqrt{(2P)^2 + Q^2} \]
\[ R' = \sqrt{4P^2 + Q^2} \]

### Step 7: Substitute Q in Terms of P and θ
From our earlier equation \( Q^2 = -2PQ \cos \theta \), we can express Q in terms of P and θ. However, we need to find the new resultant, so we will leave Q as it is for now.

### Step 8: Final Resultant Expression
Thus, the new resultant when P is doubled and is at right angles to Q is:
\[ R' = \sqrt{4P^2 + Q^2} \]

### Step 9: Conclusion
This is the expression for the new resultant when the force P is doubled and the resultant is at right angles to Q.