A force of125 N pulls due west on a body, and a second force pulls N 28.7 degrees W. The resultantforce is 212 N. Find the second force and the direction of the resultant.
To find the magnitude of the second force and the direction of the resultant, we can resolve the forces into their components.
Let's define the first force as F₁ = 125 N, pulling due west. In this case, the x-component of F₁ will be -125 N because the force is in the opposite direction of positive x-axis.
Now, let's define the second force as F₂. The given angle of N 28.7 degrees W indicates that F₂ is 28.7 degrees west of north. To find the components of F₂, we need to resolve it into its x and y components.
The x-component of F₂, F₂x, can be found using cosine function:
F₂x = F₂ * cos(28.7°)
The y-component of F₂, F₂y, can be found using sine function:
F₂y = F₂ * sin(28.7°)
Now, let's write the equation for the resultant of the two forces:
Resultant force = F₁ + F₂
Since we know the magnitude of the resultant force is 212 N, we can substitute the x and y components:
Resultant force = (-125 N) + F₂x = 212 N
0 N + F₂y = 0 N
From the second equation, we can see that F₂y must be zero since the y-component of the resultant force is zero. This means that the second force, F₂, does not have any vertical (upward or downward) component.
Solving the first equation for F₂x gives us:
F₂x = 212 N + 125 N
F₂x = 337 N
Now, let's find F₂ using the components:
F₂ = sqrt(F₂x² + F₂y²)
F₂ = sqrt((337 N)² + (0 N)²)
F₂ = sqrt(113569 N²)
F₂ ≈ 337 N
So, the magnitude of the second force, F₂, is approximately 337 N.
To find the direction of the resultant force, we can find the angle it makes with the positive x-axis using the components. Let's call this angle θ.
θ = arctan(F₂y / F₂x)
θ = arctan(0 N / 337 N)
θ = arctan(0)
θ = 0°
Therefore, the direction of the resultant force is due west, which means it is along the negative x-axis.
In summary:
- The magnitude of the second force, F₂, is approximately 337 N.
- The direction of the resultant force is due west along the negative x-axis.