a force acted upon a body by 3 given force is under equilibrium if f1 =10n and f2 =6n find f3 and angle
To find the magnitude of force f3 and the angle it makes with the other forces, we can use the method of vector addition.
The forces f1 and f2 can be represented as vectors in a two-dimensional coordinate system. Let's assume f1 acts along the positive x-axis, and f2 makes an angle θ with the positive x-axis.
Step 1: Resolve f1 and f2 into their respective x and y-components.
Since f1 is along the positive x-axis, its x-component (f1x) will be equal to its magnitude (f1) and y-component (f1y) will be zero.
f1x = f1 = 10N
f1y = 0N
To find the components of f2, we use trigonometric functions. The x-component (f2x) can be found using cosine, and the y-component (f2y) can be found using sine.
f2x = f2 * cos(θ)
f2y = f2 * sin(θ)
Step 2: Determine the x and y-components of the resultant force R.
Since the system is in equilibrium, the total x-component and total y-component of all the forces must add up to zero.
Sum of x-components = f1x + f2x + f3*x = 0
Sum of y-components = f1y + f2y + f3*y = 0
From the above equations, we can conclude that:
0 + f2 * cos(θ) + f3 * cos(φ) = 0
0 + f2 * sin(θ) + f3 * sin(φ) = 0
Step 3: Solve for f3 and φ.
Since we have two equations with two unknowns (f3 and φ), we can solve them simultaneously.
From the first equation:
f3 * cos(φ) = -f2 * cos(θ)
Dividing the first equation by the second equation:
tan(φ) = -(f2 * cos(θ)) / (f2 * sin(θ))
Simplifying:
tan(φ) = -cot(θ)
φ = atan(-cot(θ))
Once we have φ, we can substitute it back into the first equation to solve for f3:
f3 * cos(φ) = -f2 * cos(θ)
f3 = -(f2 * cos(θ)) / cos(φ)
Substitute the given values:
f2 = 6N
θ is the given angle
Solve for φ using the equation φ = atan(-cot(θ))
Then substitute the values of f2, θ, and φ in f3 = -(f2 * cos(θ)) / cos(φ) to find f3.
Note: Remember to convert the angle to radians if necessary before performing trigonometric calculations.
F3 =
-16 N. = 16N.[180o].