Two forces 5N and 4N are inclined to each other at 30°.Find the resultant force?

FreshTheo

2 years ago

To find the resultant force and direction using the triangle method, we can draw a triangle where the two given forces represent two sides and the resultant force represents the third side. The angle between the two given forces is also given, which is 30°.

Here are the steps to find the resultant force and direction using the triangle method:

Draw a triangle with two sides representing the magnitudes of the two given forces, 5N and 4N, and the included angle between them, which is 30°. Label the sides and the angle appropriately.
To find the magnitude of the resultant force, we can use the Law of Cosines, which states that the square of the length of the third side of a triangle (c^2) is equal to the sum of the squares of the other two sides (a^2 and b^2) minus twice the product of the lengths of those sides (2ab) times the cosine of the included angle (θ).
In this case, we want to find the magnitude of the resultant force (c), so we can write:

c^2 = a^2 + b^2 - 2ab cos(θ)

Substituting the values we have:

c^2 = 5^2 + 4^2 - 2(5)(4) cos(30°)

c^2 = 25 + 16 - 40 cos(30°)

c^2 = 25 + 16 - 40(√3/2)

c^2 = 41 - 20√3

c ≈ 2.23 N

Therefore, the magnitude of the resultant force is approximately 2.23 N.

To find the direction of the resultant force, we can use the Law of Sines, which states that the ratio of the length of each side of a triangle to the sine of the opposite angle is the same for all three sides.
In this case, we want to find the angle between the resultant force and the 5N force. Let's call this angle α. We can write:

sin(α)/c = sin(30°)/5

Solving for sin(α):

sin(α) = c sin(30°)/5

Substituting the value of c we found:

sin(α) = 2.23 sin(30°)/5

sin(α) ≈ 0.38

Taking the inverse sine:

α ≈ 22.1°

Therefore, the direction of the resultant force with respect to the 5N force is approximately 22.1°.