The greatest and least resultant of two forces acting at a point is 29 kgwt and 5 kgwt respectively . If each force is increased by 3 kgwt , find the resultant of the two new forces when acting at right angles to each other ?

Explanation

To solve the problem step by step, we will follow the information given and use the properties of vectors.

### Step 1: Define the Forces
Let the two forces be \( P \) and \( Q \). According to the problem, we know:
- The greatest resultant \( R_{max} = P + Q = 29 \, \text{kgwt} \)
- The least resultant \( R_{min} = P - Q = 5 \, \text{kgwt} \)

### Step 2: Set Up the Equations
From the information provided, we can set up the following equations:
1. \( P + Q = 29 \) (Equation 1)
2. \( P - Q = 5 \) (Equation 2)

### Step 3: Solve the Equations
To find the values of \( P \) and \( Q \), we can add the two equations:
\[ (P + Q) + (P - Q) = 29 + 5 \]
This simplifies to:
\[ 2P = 34 \implies P = \frac{34}{2} = 17 \, \text{kgwt} \]

Next, substitute \( P \) back into Equation 1 to find \( Q \):
\[ 17 + Q = 29 \implies Q = 29 - 17 = 12 \, \text{kgwt} \]

### Step 4: Increase the Forces
Now, we increase each force by \( 3 \, \text{kgwt} \):
- New force \( P' = P + 3 = 17 + 3 = 20 \, \text{kgwt} \)
- New force \( Q' = Q + 3 = 12 + 3 = 15 \, \text{kgwt} \)

### Step 5: Calculate the Resultant of the New Forces
Since the new forces are acting at right angles to each other, we can find the resultant \( R' \) using the Pythagorean theorem:
\[ R' = \sqrt{(P')^2 + (Q')^2} \]
Substituting the values:
\[ R' = \sqrt{(20)^2 + (15)^2} = \sqrt{400 + 225} = \sqrt{625} = 25 \, \text{kgwt} \]

### Final Answer
The resultant of the two new forces when acting at right angles to each other is \( 25 \, \text{kgwt} \).

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