1. Consider four vectors ~F1, ~F2, ~F3, and ~F4, where their magnitudes are F1= 57 N, F2= 38 N, F3= 25 N, and F4= 67 N. Let θ1= 130◦, θ2=−130◦, θ3= 29◦, and θ4=−64◦, measured from the positive x axis
with the counter-clockwise angular direction as positive. What is the magnitude of the resultant vector ~F, where ~F=~F1+~F2+~F3+~F4?
Answer in units of N.
2.What is the direction of this resultant vector ~F?
Note:
Give the angle in degrees, use counterclockwise as the positive angular direction, between the limits of −180◦ and +180◦ from the positive x axis.
Answer in units of◦
To find the magnitude of the resultant vector ~F (~F1 + ~F2 + ~F3 + ~F4), we can use the Pythagorean theorem. The Pythagorean theorem states that for a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
1. Calculate the x-components and y-components of each vector using trigonometry:
- For vector ~F1:
x-component = F1 * cos(θ1)
y-component = F1 * sin(θ1)
- For vector ~F2:
x-component = F2 * cos(θ2)
y-component = F2 * sin(θ2)
- For vector ~F3:
x-component = F3 * cos(θ3)
y-component = F3 * sin(θ3)
- For vector ~F4:
x-component = F4 * cos(θ4)
y-component = F4 * sin(θ4)
2. Calculate the sum of the x-components and the sum of the y-components:
- Sum of x-components = x-component of ~F1 + x-component of ~F2 + x-component of ~F3 + x-component of ~F4
- Sum of y-components = y-component of ~F1 + y-component of ~F2 + y-component of ~F3 + y-component of ~F4
3. Apply the Pythagorean theorem to find the magnitude of ~F:
- Magnitude of ~F = sqrt((Sum of x-components)^2 + (Sum of y-components)^2)
To find the direction of ~F, you can use the inverse tangent function (arctan) to find the angle between ~F and the positive x-axis.
4. Calculate the angle using the arctan function:
- Angle = arctan((Sum of y-components) / (Sum of x-components))
Now, let's calculate the magnitude and direction of the resultant vector ~F.
1. Calculate the x and y components of each vector using the given magnitudes and angles:
- For vector ~F1: x-component = 57 * cos(130°), y-component = 57 * sin(130°)
- For vector ~F2: x-component = 38 * cos(-130°), y-component = 38 * sin(-130°)
- For vector ~F3: x-component = 25 * cos(29°), y-component = 25 * sin(29°)
- For vector ~F4: x-component = 67 * cos(-64°), y-component = 67 * sin(-64°)
2. Calculate the sum of the x-components and y-components:
- Sum of x-components = 57 * cos(130°) + 38 * cos(-130°) + 25 * cos(29°) + 67 * cos(-64°)
- Sum of y-components = 57 * sin(130°) + 38 * sin(-130°) + 25 * sin(29°) + 67 * sin(-64°)
3. Apply the Pythagorean theorem to find the magnitude of ~F:
- Magnitude of ~F = sqrt((Sum of x-components)^2 + (Sum of y-components)^2)
4. Calculate the direction of ~F using the arctan function:
- Angle = arctan((Sum of y-components) / (Sum of x-components))
Now you can substitute the actual values and solve for the magnitude and direction of the resultant vector ~F.
I showed you how to get the magnitude. Scroll down.
tan theta = sumy/sumx