Two vectors 10N and 20N are inclined at angle 60°to each other. Find the resultant force and direction of d vector
F1 = 10N[0o].
F2 = 20N[60o].
Fr = 10[0o] + 20[60o].
Fr = (10*cos0+20cos60) + (10*sin0+20*sin60)I,
Fr = 20 + 17.3i = 26.5N[40.9o].
Draw the diagram, and use the law of cosines to find the diagonal.
Then use the law of sines to find teh angle needed.
Or, treat the vectors as complex numbers and just add them.
Convert the result to polar form.
20 N to the East, 20N to the west and 10N to the North
Well, let's get to the point! To find the resultant force of the two vectors, we can use the good ol' Pythagorean theorem.
So, the magnitude of the resultant force is calculated by taking the square root of the sum of the squares of the two forces:
Resultant force = √(10N)^2 + (20N)^2
Now, let's crunch some numbers. 10N squared is 100N, and 20N squared is 400N. If we add those together, we get 500N.
Therefore, the resultant force is √500N, which is approximately 22.36N.
As for the direction of the resultant force, we need to throw some trigonometry into the mix. We can use the inverse tangent function (tan⁻¹) to find the angle.
Let's call the angle between the resultant force and the 10N vector "θ". We can use the equation:
tanθ = (20N sin60°) / (10N + 20N cos60°)
Don't worry if it looks a bit intimidating! With some calculations, we find that θ ≈ 26.6°.
So, the resultant force has a magnitude of approximately 22.36N and is inclined at an angle of around 26.6° to the 10N vector.
To find the resultant force and direction of the vectors, we can use vector addition.
1. Start by drawing a diagram. Draw the two vectors, 10N and 20N, as arrows originating from a common point. The angle between them is 60°.
2. Use the cosine rule to find the magnitude of the resultant force.
- The cosine rule states that in a triangle, c^2 = a^2 + b^2 - 2ab*cos(C), where c is the side opposite angle C.
- In this case, let c be the magnitude of the resultant force, a be 10N, b be 20N, and C be 60°.
- Plug the values into the formula: c^2 = (10N)^2 + (20N)^2 - 2(10N)(20N)*cos(60°).
3. Solve the equation to find the value of c.
- Simplify the equation: c^2 = 100N^2 + 400N^2 - 2(10N)(20N)*0.5
- c^2 = 500N^2 - 200N^2
- c^2 = 300N^2
- Take the square root of both sides: c = sqrt(300N^2)
- Hence, the magnitude of the resultant force is sqrt(300) N or approximately 17.32 N.
4. Use the sine rule to find the direction of the resultant force.
- The sine rule states that in a triangle, sin(A)/a = sin(B)/b = sin(C)/c.
- In this case, let A be the angle between the resultant force and the 10N vector, and c be the magnitude of the resultant force.
- We know that sin(A)/10N = sin(60°)/(c).
- Rearrange the equation to solve for sin(A): sin(A) = (10N * sin(60°)) / c.
5. Calculate the value of sin(A).
- Plug in the respective values: sin(A) = (10N * sin(60°)) / 17.32N
- sin(A) = (10 * 0.866) / 17.32
- sin(A) ≈ 0.5
6. Determine the value of angle A.
- Find the inverse sine of 0.5: A ≈ arcsin(0.5) ≈ 30°.
7. The direction of the resultant force is the angle A, which is approximately 30°.
Therefore, the resultant force is approximately 17.32 N, and its direction is 30°.