Two vectors 5n and 8n act at an angle 45 calculate the resultant
Say 8 in x direction and 5 up 45 degrees from x axis direction
X direction = 8 + 5 * cos 45 = 8 + 3.54 = 11.5
Y direction = 5 sin 45 = 3.54
magnitude = sqrt (11.5^2 + 3.54^2)
angle above x axis = tan^-1 ( 3.54 / 11.5)
Well, let me put it this way: when an angle of 45 degrees is involved, things can get a bit "acute"!
To find the resultant of the two vectors, we need to use a little "trig"! Specifically, we can use the cosine rule, which states that the square of the resultant equals the sum of the squares of the individual vectors, minus twice the product of their magnitudes and the cosine of the angle between them.
Now, if we let the magnitude of the first vector be 5n and the magnitude of the second vector be 8n, we can plug these values into the formula. So, the resultant squared would be:
(5n)^2 + (8n)^2 - 2 * 5n * 8n * cos(45)
If we simplify this, we get:
25n^2 + 64n^2 - 80n^2 * cos(45)
Combining like terms, we have:
89n^2 - 80n^2 * cos(45)
And if we evaluate this expression further, we get:
(89 - 80 * cos(45))n^2
Now, cos(45) is equal to (√2)/2. Let me calculate this real quick...
So, plugging that value back into the expression, we have:
(89 - 80 * (√2)/2)n^2
And if we simplify it even further, we get:
(89 - 40√2)n^2
So, the resultant would be approximately (89 - 40√2)n^2.
To calculate the resultant of two vectors, we can use the head-to-tail method or the parallelogram method. In this case, we can use the head-to-tail method.
Step 1: Draw the vectors
Draw a vector with a magnitude of 5n and label it as vector A. Then draw another vector with a magnitude of 8n at an angle of 45 degrees from vector A. Label this vector as vector B.
Step 2: Find the components of the vectors
Since the vectors are not in the x-y coordinate system, we need to find their components. We can do this by using basic trigonometry.
- For vector A:
- A = 5n * cos(0) = 5n
- A' = 5n * sin(0) = 0
- For vector B:
- B = 8n * cos(45) ≈ 5.66n
- B' = 8n * sin(45) ≈ 5.66n
Step 3: Add the components
Add the x-component (A + B) and the y-component (A' + B') to get the resultant vector.
- X-component: A + B = 5n + 5.66n = 10.66n
- Y-component: A' + B' = 0 + 5.66n = 5.66n
So the resultant vector is approximately 10.66n in the x-direction and 5.66n in the y-direction.
To calculate the resultant of two vectors when they act at an angle, you can use the cosine law or the component method. I will explain both methods, and you can choose the one that you feel more comfortable with.
Method 1: Cosine Law
1. Draw a diagram representing the vectors. Label the vectors as A = 5N and B = 8N, and the angle between them as θ = 45 degrees.
2. Apply the cosine law formula:
c² = a² + b² - 2ab * cos(θ)
where c is the magnitude of the resultant vector, and a and b are the magnitudes of the given vectors.
In this case, the formula becomes:
R² = 5² + 8² - 2 * 5 * 8 * cos(45)
R² = 25 + 64 - 2 * 5 * 8 * cos(45)
R² = 89 - 80 * cos(45)
3. Evaluate the value of R:
R = √(89 - 80 * cos(45))
Method 2: Component Method
1. Draw a diagram representing the vectors. Label the vectors as A = 5N and B = 8N, and the angle between them as θ = 45 degrees.
2. Resolve each vector into its x and y components. The x-components can be found using the formula Ax = A * cos(θ), and the y-components can be found using the formula Ay = A * sin(θ).
Ax = 5 * cos(45) = 5 * (√2 / 2) = 5 * (√2) / 2 = 5√2 / 2
Ay = 5 * sin(45) = 5 * (√2 / 2) = 5 * (√2) / 2 = 5√2 / 2
Bx = 8 * cos(45) = 8 * (√2 / 2) = 8 * (√2) / 2 = 8√2 / 2
By = 8 * sin(45) = 8 * (√2 / 2) = 8 * (√2) / 2 = 8√2 / 2
3. Add the x-components and y-components separately to find the resultant vector.
Rx = Ax + Bx = 5√2 / 2 + 8√2 / 2 = (5 + 8)√2 / 2 = (13√2) / 2
Ry = Ay + By = 5√2 / 2 + 8√2 / 2 = (5 + 8)√2 / 2 = (13√2) / 2
4. Calculate the magnitude of the resultant vector using the Pythagorean theorem:
R = √(Rx² + Ry²)
R = √(((13√2) / 2)² + ((13√2) / 2)²)
R = √((169/4)∗2+(169/4)∗2)
R = √(338/2+338/2)
R = √(338+338)/2
R = √676/2
R = 26/√2
Therefore, the magnitude of the resultant vector (R) is 26/√2.