The angle between two forces of magnitude 7N and 4N is 120°. Find in N, the magnitude of their resultant.

Question

ScanSolve AI Tutor · Accepted Answer

To find the magnitude of the resultant force, we use the Law of Cosines for forces.

Identify the given magnitudes of the forces and the angle between them.
Let $F_1 = 7 \, \text{N}$
Let $F_2 = 4 \, \text{N}$
Let $\theta = 120^\circ$
Apply the formula for the magnitude of the resultant force ( $R$ ).
The formula is:
$R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2 \cos \theta}$
Substitute the given values into the formula.
$R = \sqrt{(7 \, \text{N})^2 + (4 \, \text{N})^2 + 2(7 \, \text{N})(4 \, \text{N}) \cos(120^\circ)}$
Calculate the square of the forces and the cosine of the angle.
We know that $\cos(120^\circ) = -\frac{1}{2}$ .
$R = \sqrt{49 \, \text{N}^2 + 16 \, \text{N}^2 + 2(28 \, \text{N}^2) \left(-\frac{1}{2}\right)}$
Perform the arithmetic operations.
$R = \sqrt{49 \, \text{N}^2 + 16 \, \text{N}^2 - 28 \, \text{N}^2}$ $R = \sqrt{65 \, \text{N}^2 - 28 \, \text{N}^2}$ $R = \sqrt{37 \, \text{N}^2}$
Calculate the final magnitude.
$R = \sqrt{37} \, \text{N}$
The magnitude of their resultant is $\sqrt{37} \, \text{N}$ .

Answer