Two forces of mangitude `F_(1)` and `F_(2)` are acting on an object.The magnitude of the net force `F_(n e t )` on the object will be in the range.

Options

A. `F_(1) le F_("net") le F_(2)`
B. `|F_(1) - F_(2)| //2le F_("net") le (F_(1) + F_(2) ) //2`
C. `|F_(1) -F_(2) | le F_("net") le F_(1) + F_(2)`
D. `|F_(1) - F_(2) |ge F_("net") | F_(1) + F_(2) |`

Answer

To determine the range of the net force \( F_{\text{net}} \) acting on an object due to two forces \( F_1 \) and \( F_2 \), we can use the principles of vector addition. The net force can vary depending on the angle between the two forces. Here’s how we can find the minimum and maximum values of \( F_{\text{net}} \):

### Step 1: Understand the Forces
We have two forces acting on an object:
- Force \( F_1 \)
- Force \( F_2 \)

### Step 2: Use the Resultant Force Formula
The magnitude of the resultant force \( F_{\text{net}} \) when two forces are acting at an angle \( \theta \) can be calculated using the formula:
\[ F_{\text{net}} = \sqrt{F_1^2 + F_2^2 + 2F_1F_2 \cos \theta} \]

### Step 3: Determine Maximum Value of \( F_{\text{net}} \)
To find the maximum value of \( F_{\text{net}} \), we need to maximize the term \( \cos \theta \). The maximum value of \( \cos \theta \) is 1 (which occurs when the forces are in the same direction). Thus:
\[ F_{\text{net, max}} = \sqrt{F_1^2 + F_2^2 + 2F_1F_2 \cdot 1} = \sqrt{(F_1 + F_2)^2} = F_1 + F_2 \]

### Step 4: Determine Minimum Value of \( F_{\text{net}} \)
To find the minimum value of \( F_{\text{net}} \), we need to minimize the term \( \cos \theta \). The minimum value of \( \cos \theta \) is -1 (which occurs when the forces are in opposite directions). Thus:
\[ F_{\text{net, min}} = \sqrt{F_1^2 + F_2^2 + 2F_1F_2 \cdot (-1)} = \sqrt{F_1^2 + F_2^2 - 2F_1F_2} = \sqrt{(F_1 - F_2)^2} = |F_1 - F_2| \]

### Step 5: Conclusion
The magnitude of the net force \( F_{\text{net}} \) will be in the range:
\[ |F_1 - F_2| \leq F_{\text{net}} \leq F_1 + F_2 \]