Question

As you will see in a later chapter, forces are vector quantities, and the total force on an object is the vector sum of all forces acting on it.In the figure below, a force F1 of magnitude 5.90 units acts on a box at the origin in a direction = 31.0° above the positive x-axis. A second force F2 of magnitude 5.00 units acts on the box in the direction of the positive y-axis. Find graphically the magnitude and direction (in degrees counterclockwise from the +x-axis) of the resultant force F1 + F2.Two forces act on a box. Force vector F1 acts up and right on the right side of the box at an acute angle above the horizontal. Force vector F2 acts vertically upwards on the top side of the box.

Answer 1

We are asked to determine the sum of the two vectors F1 and F2. To do that we will use the following triangle:

Therefore, substituting the magnitudes of the vectors the triangle is:

Now, to determine the magnitude of the resultant force "R" we will use the cosine law:

$c^2=a^2+b^2-2ab\cos\theta$

Where:

$\begin{gathered} c=\text{ opposite side to the 120\degree angle} \\ a,b=\text{ other sides of the triangle} \end{gathered}$

Now, we substitute the values:

$R^2=5^2+5.9^2-2(5)(5.9)\cos(121)$

Solving the operations we get:

$R^2=90.25$

Now, we take the square root to both sides:

$\begin{gathered} R=\sqrt{90.25} \\ R=9.5 \end{gathered}$

Therefore, the magnitude of the sum of vectors is 9.5

Now, we determine the angle:

We need to determine angle "y". To do that we will determine the angle "x" using the sine law:

$\frac{\sin A}{a}=\frac{\sin B}{b}$

Where:

$\begin{gathered} a=\text{ side opposite to angle A} \\ b=\text{ ide oppoite to angle B} \end{gathered}$

Now, we plug in the known values:

$\frac{\sin x}{5}=\frac{\sin121}{9.5}$

Now, we multiply both sides by 5:

$\sin x=5\frac{\sin121}{9.5}$

Solving the operations:

$\sin x=0.45$

Now, we take the inverse function of the sine:

$x=\sin^{-1}(0.45)$

Solving the operations:

$x=26.83$

Now, we have that since F1 is a vertical force then the sum of angles "x" and "y" must add up to 90:

$x+y=90$

Now, we substitute the value of angle "x":

$26.83+y=90$

Now, we subtract 26.83 from both sides:

$\begin{gathered} y=90-26.83 \\ y=63.17 \end{gathered}$

Therefore, the angle of the sum of vectors is 63.17°.