a force of 200N is inclined at an angle 120° to another force P. The angle between the 200N force and the resultant force is 50°. Find the magnitude of (a) force P (b) resultant of the two forces

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It is easy to understand

To find the magnitude of force P and the resultant of the two forces, we can use vector addition and trigonometry.

(a) Finding the magnitude of force P:
Let's assume the magnitude of force P is denoted by "P" (without units).

Using the trigonometric definition of a parallelogram of forces, we can create a parallelogram using the given 200N force and force P. The angle between these two forces is given as 120°.

To find the magnitude of force P, we need to find the horizontal component of the parallelogram, which is adjacent to the angle formed by the 200N force and force P.

To find the horizontal component of force P, we can use the cosine rule. The formula for the cosine rule is:

c^2 = a^2 + b^2 - 2ab * cos(C)

In this case, let's assume the 200N force is side "a" and force P is side "b," with the angle between them being 120° (angle C). We want to find side "b" (force P).

Plugging in the given values:
200N^2 = P^2 + 200N^2 - 2(P)(200N) * cos(120°)

Simplifying the equation, we have:
200N^2 = P^2 + 40000N^2 - 400(P)N * (-0.5)

200N^2 = P^2 + 40000N^2 + 200PN

Rearranging the equation:
P^2 + 200PN - 200N^2 = 0

Now we have a quadratic equation. We can solve it using the quadratic formula:

P = (-b ± √(b^2 - 4ac)) / (2a)

Solving the quadratic equation, we find two possible values for P. However, since we're dealing with forces (which are positive by convention), we can ignore the negative value. Therefore, the magnitude of force P is:

P ≈ 126.76N

(b) Finding the magnitude of the resultant of the two forces:
To find the magnitude of the resultant force, we can use the cosine rule again. This time, we need to find side "c" (resultant force).

Using the cosine rule:
c^2 = a^2 + b^2 - 2ab * cos(C)

Let's assume the 200N force is side "a," force P is side "b," and the angle between them is given as 50° (angle C).

Plugging in the given values:
c^2 = 200N^2 + (126.76N)^2 - 2(200N)(126.76N) * cos(50°)

c^2 = 40000N^2 + 16057.8576N^2 - 2(200)(126.76N) * 0.642788

c^2 = 56057.8576N^2 - 51255.344N + (2)(200)(126.76N) * 0.642788

Simplifying the equation:
c^2 ≈ 1606.97631N^2 - 51255.344N + 21482.6421776N

To find the magnitude of the resultant force, we take the square root of c^2:

c ≈ √(1606.97631N^2 - 51255.344N + 21482.6421776N)

c ≈ √(43589.61883N^2 - 51255.344N)

c ≈ √(30766.72583N^2)

c ≈ 175.42N

Therefore, the magnitude of the resultant of the two forces is approximately 175.42N.