Question

two forces act at a point in the plane. the angle between the two forces is given. find the magnitude of the resultant force.
forces of 348 and 340 newtons, forming an angle of 63°

the magnitude of the resultant force is \boxed{} newtons.
(round to the nearest newton as needed.)

Explanation:

Step1: Recall the Law of Cosines for Resultant Force

When two forces \( F_1 \) and \( F_2 \) act at an angle \( \theta \) between them, the magnitude of the resultant force \( R \) is given by the formula:

$$ R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2\cos\theta} $$

Here, \( F_1 = 348 \) N, \( F_2 = 340 \) N, and \( \theta = 63^\circ \).

Step2: Substitute the Values into the Formula

First, calculate \( F_1^2 \), \( F_2^2 \), and \( 2F_1F_2\cos\theta \):

• \( F_1^2 = 348^2 = 348\times348 = 121104 \)
• \( F_2^2 = 340^2 = 340\times340 = 115600 \)
• Calculate \( 2F_1F_2\cos\theta \):
• \( 2\times348\times340 = 2\times118320 = 236640 \)
• \( \cos(63^\circ) \approx 0.4540 \)
• So, \( 2F_1F_2\cos\theta \approx 236640\times0.4540 \approx 107434.56 \)

Step3: Sum the Terms Inside the Square Root

Add \( F_1^2 \), \( F_2^2 \), and \( 2F_1F_2\cos\theta \):

$$ 121104 + 115600 + 107434.56 = 121104 + 115600 = 236704; \quad 236704 + 107434.56 = 344138.56 $$

Step4: Take the Square Root

Now, find the square root of \( 344138.56 \):

$$ R = \sqrt{344138.56} \approx 586.62 $$

Rounding to the nearest newton, we get \( R \approx 587 \).

Answer:

\( \boxed{587} \)