\[ F_a = 4000\ N \]

**– The angle between Fa and line L is $\theta_a = 45^{\circ}$.**

**– The angle between Fb and line L is $\theta_b = 35^{\circ}$.**

The question aims to find the **2nd force** exerted on the **housing unit** by a snowcat in Antarctica, and the sum of both forces’ **magnitude** exerted on the **housing unit.**

The question depends on the concept of **Force,** and **two forces** exerted on an **object** at an **angle,** and the **resultant force.** The **force** is a **vector** quantity; thus, it has a **direction** along with the **magnitude.** The **resultant force** is the **vector sum** of two forces acting on an object at different **angles.** The **resultant force** is given as:

\[ \overrightarrow{R} = \overrightarrow{F_1} + \overrightarrow{F_2} \]

## Expert Answer

The **sum** of **forces** exerted by the **snowcats** on the housing unit is **parallel** to **line L**. This means that the **forces** must be balanced in the **horizontal component.** The **balanced equation** of the **horizontal components** of these **forces** is given as:

\[ F_a \cos \theta_a = F_b \cos \theta_b \]

Substituting the values, we get:

\[ 4000 \cos (45 ^{\circ}) = F_b \cos (35^ {\circ}) \]

Rearranging for $F_b$, we get:

\[ F_b = \dfrac{ 4000 \cos( 45^{\circ}) }{ \cos ( 35^{\circ} } \]

\[ F_b = \dfrac{ 4000 \times 0.707 }{ 0.819 } \]

\[ F_b = \dfrac{ 2828 }{ 0.819 } \]

\[ F_b = 3453\ N \]

The sum of both **forces** $F_a$ and $F_b$ is given as:

\[ \overrightarrow{F}^2 = \overrightarrow{F_a}^2 + \overrightarrow{F_b}^2 \]

The **magnitude** of $F_a$ is given as:

\[ F_a = 4000 \sin (45) \]

\[ F_a = 4000 \times 0.707 \]

\[ F_a = 2828\ N \]

The **magnitude** of $F_b$ is given as:

\[ F_b = 3453 \sin (35) \]

\[ F_b = 3453 \times 0.5736 \]

\[ F_b = 1981\ N \]

The **sum** of the **magnitude** of both forces is given as:

\[ F = \sqrt{ F_a^2 + F_b^2 } \]

Substituting the values, we get:

\[ F = \sqrt{ 2828^2 + 1981^2 } \]

\[ F = 3453\ N \]

## Numerical Result

The **magnitude** of $F_b$ is calculated to be:

\[ F_b = 3453\ N \]

The **magnitude** of the **sum** of both **forces** is calculated to be:

\[ F = 3453\ N \]

## Example

Two **forces, 10N** and **15N,** are exerted on an object at an angle of **45.** Find the **resultant force** on the object.

\[ F_a = 10\ N \]

\[ F_b = 15\ N \]

\[ \theta = 45^ {\circ} \]

The **resultant force** between these two forces is given as:

\[ F = \sqrt{ |F_a|^2 + |F_b|^2 } \]

The **magnitude** of $F_a$ is given as:

\[ F_a = 10 \sin (45) \]

\[ F_a = 10 \times 0.707 \]

\[ F_a = 7.07\ N \]

The **magnitude** of $F_b$ is given as:

\[ F_b = 15 \sin (45) \]

\[ F_b = 15 \times 0.707 \]

\[ F_b = 10.6\ N \]

The **resultant force** is given as:

\[ F = \sqrt{ 7.07^2 + 10.6^2 } \]

\[ F = \sqrt{ 49.98 + 112.36 } \]

\[ F = \sqrt{ 162.34 } \]

\[ F = 12.74\ N \]