To pull an old stump out of the ground, you and a friend tie two ropes to the stump. You pull on it with a force of 500 N to the north while your friend pulls with a force of 450 N to the northwest. The total force from the two ropes is

(a) less than 950 N.

(b) exactly 950 N.

(c) more than 950 N.

Resultant force can be obtained when different forces are acting on the system.

It is a single force and has a similar effect like the other forces when they are combined.

The magnitude of the resultant force can be evaluated with the help of the parallelogram law.

The forces can be shown as:

Here, the x-axis is considered as the W-E direction. The y-axis can be considered as the N-S direction. The angle of inclination of the force in the northwest direction is taken to be 45 degrees.The pulling forces in the ropes can be shown outward, as depicted in the above figure.

The given data can be listed below as:

• The force acting in the north direction is $F_1=500\mathrm{N}$.
• The force acting in the northwest direction is $F_2=450\mathrm{N}$.
• The angle of inclination of $F_2$force with the y-axis is $\theta =45°$.

From the parallelogram law, the resultant force from the two ropes can be calculated as:

$R=\sqrt{F_1^2+F_1^2+2F_1{F}_2\cos \theta }$

Substitute the values as 500 N for $F_1$, 450 N for $F_2$, and $45°$for $\theta$in the above equation.

$\begin{array}{c}R=\sqrt{{\left(500\mathrm{N}\right)}^2+{\left(450\mathrm{N}\right)}^2+2\times 500\mathrm{N}\times 450\mathrm{N}\times \cos 45°}\\ =\sqrt{770698.0515}\mathrm{N}\\ =877.894\mathrm{N}\end{array}$

The resultant force is $877.894\mathrm{N}$.

Option (a) is correct because the resultant force is $877.894\mathrm{N}$, which is less than 950 N.

Option (b) is incorrect because the resultant force is not equal to 950 N.

Option (c) is incorrect because the resultant force is not more than 950 N.

Thus, from the above analysis of the options, it can be concluded that option (a), which is less than 950 N, is correct.