To solve the problem step by step, we will follow the reasoning and calculations presented in the video transcript.
### Step 1: Identify the Forces
We are given two forces:
- The greater force \( F_1 = 7 \, \text{N} \)
- The lesser force \( F_2 = 3 \, \text{N} \)
### Step 2: Set Up the Equations for Resultants
The maximum resultant occurs when the angle between the two forces is \( 0^\circ \):
\[
R_{\text{max}} = F_1 + F_2
\]
The minimum resultant occurs when the angle is \( 180^\circ \):
\[
R_{\text{min}} = |F_1 - F_2|
\]
From the problem, we know:
\[
R_{\text{max}} = 7 \, \text{N} \quad \text{and} \quad R_{\text{min}} = 3 \, \text{N}
\]
### Step 3: Set Up the Equations
From the maximum resultant:
\[
F_1 + F_2 = 7 \quad \text{(1)}
\]
From the minimum resultant:
\[
|F_1 - F_2| = 3 \quad \text{(2)}
\]
### Step 4: Solve the Equations
From equation (1):
\[
F_1 + F_2 = 7
\]
From equation (2), we can have two cases:
1. \( F_1 - F_2 = 3 \)
2. \( F_2 - F_1 = 3 \)
#### Case 1: \( F_1 - F_2 = 3 \)
Adding equations (1) and (2):
\[
(F_1 + F_2) + (F_1 - F_2) = 7 + 3
\]
\[
2F_1 = 10 \implies F_1 = 5 \, \text{N}
\]
Substituting \( F_1 \) back into equation (1):
\[
5 + F_2 = 7 \implies F_2 = 2 \, \text{N}
\]
#### Case 2: \( F_2 - F_1 = 3 \)
Adding equations (1) and (2):
\[
(F_1 + F_2) + (F_2 - F_1) = 7 + 3
\]
\[
2F_2 = 10 \implies F_2 = 5 \, \text{N}
\]
Substituting \( F_2 \) back into equation (1):
\[
F_1 + 5 = 7 \implies F_1 = 2 \, \text{N}
\]
Thus, we have:
- \( F_1 = 5 \, \text{N} \)
- \( F_2 = 2 \, \text{N} \)
### Step 5: Increase Each Force by 3 N
Now, we increase both forces by \( 3 \, \text{N} \):
\[
F_1' = 5 + 3 = 8 \, \text{N}
\]
\[
F_2' = 2 + 3 = 5 \, \text{N}
\]
### Step 6: Calculate the Resultant Force
The angle between the two forces is \( 60^\circ \). The formula for the resultant \( R \) of two forces is:
\[
R = \sqrt{F_1'^2 + F_2'^2 + 2 F_1' F_2' \cos(\theta)}
\]
Substituting the values:
\[
R = \sqrt{8^2 + 5^2 + 2 \cdot 8 \cdot 5 \cdot \cos(60^\circ)}
\]
Since \( \cos(60^\circ) = \frac{1}{2} \):
\[
R = \sqrt{64 + 25 + 2 \cdot 8 \cdot 5 \cdot \frac{1}{2}}
\]
\[
= \sqrt{64 + 25 + 40}
\]
\[
= \sqrt{129}
\]
### Final Answer
The magnitude of the resultant force is:
\[
R = \sqrt{129} \, \text{N}
\]
