Find the magnitude of the resultant force and the angle it makes with the positive x-axis. (Let |a| = 30 lb and |b| = 16 lb. Round your answers to one decimal place.)
lal point upward as a 45 degree angle on the first quadrant
lbl point downward as a 30 degree angle on the fourth quadrant
X = 30*Cos(45) + 16*Cos(-30)
Y= 30*Sin(45) + 16*Sin(-30)
Fnet = Sqrt(X^2 +Y^2)
hope that helps, still working on the angle
To find the magnitude of the resultant force, we need to find the sum of the magnitudes of vectors a and b. Given that |a| = 30 lb and |b| = 16 lb, the magnitude of the resultant force (|R|) can be calculated using the Pythagorean theorem:
|R|^2 = |a|^2 + |b|^2
|R|^2 = (30 lb)^2 + (16 lb)^2
|R|^2 = 900 lb^2 + 256 lb^2
|R|^2 = 1156 lb^2
Taking the square root of both sides, we find that:
|R| = √1156 lb
|R| ≈ 34 lb (rounded to one decimal place)
So, the magnitude of the resultant force is approximately 34 lb.
To find the angle that the resultant force makes with the positive x-axis, we can draw a right triangle with sides representing the x and y components of the resultant force.
The x-component of the resultant force is given by:
Rx = |R| * cosθ
Since the angle θ is formed by vector a with the positive x-axis and is given as 45 degrees, we can substitute the values into the equation:
Rx = (34 lb) * cos(45°)
Using the cosine of 45 degrees (which is √2/2), we get:
Rx = (34 lb) * (√2/2)
Rx = 17√2 lb
The y-component of the resultant force is given by:
Ry = |R| * sinθ
Since the angle θ is formed by vector a with the positive x-axis and is given as 45 degrees, we can substitute the values into the equation:
Ry = (34 lb) * sin(45°)
Using the sine of 45 degrees (which is √2/2), we get:
Ry = (34 lb) * (√2/2)
R y = 17√2 lb
Now, we can find the angle that the resultant force makes with the positive x-axis using the arctangent function:
θ = arctan(Ry / Rx)
θ = arctan((17√2 lb) / (17√2 lb))
θ = arctan(1)
θ = 45°
So, the angle that the resultant force makes with the positive x-axis is 45 degrees.
To find the magnitude of the resultant force, we can use the formula for the magnitude of the resultant vector when two vectors are added:
|R| = sqrt(|A|^2 + |B|^2 + 2 * |A| * |B| * cos(theta))
Where |A| and |B| are the magnitudes of the vectors, and theta is the angle between them.
In this case, |A| = 30 lb and |B| = 16 lb.
To calculate the angle between the vectors, we can use the angle sum formula for two vectors:
tan(theta) = (|Ay| + |By|) / (|Ax| + |Bx|)
From the given information, we know that |Ax| = |Bx| = 0, since both vectors only have y-components. We also know that |Ay| = 30 lb and |By| = -16 lb.
Plugging in the values, we have:
tan(theta) = (30 lb + (-16 lb)) / (0 + 0) = 14 lb / 0 lb
Since the denominator is zero, we can conclude that the angle theta is 90 degrees.
Now we can use the formula for the magnitude of the resultant force:
|R| = sqrt((30 lb)^2 + (16 lb)^2 + 2 * 30 lb * 16 lb * cos(90))
Simplifying further:
|R| = sqrt(900 lb^2 + 256 lb^2 + 0 lb^2) = sqrt(1156 lb^2)
Calculating the square root:
|R| ≈ 34 lb (rounded to one decimal place)
To determine the angle the resultant force makes with the positive x-axis, we can see that the resultant force will be in the second quadrant since both vectors have components in the positive y-direction. Therefore, the angle with the positive x-axis will be 180 degrees plus the angle theta we calculated earlier.
Angle with positive x-axis = 180 degrees + 90 degrees = 270 degrees
So, the magnitude of the resultant force is approximately 34 lb and the angle it makes with the positive x-axis is 270 degrees.
All angles are measured CCW from +x-axis.
Fr = 30Lb[45o] + 16Lb[330o]
X = 30*Cos45 + 16*Cos330 = 35.1 Lbs.
Y = 30*sin45 + 16*sin330 = 13.2 Lbs.
Fr = 35.1 + 13.2i = 37.5Lbs[21o] = Resultant force and direction.