(c) A force vector with a magnitude of 44 lb that is in the third quadrant with an x component whose magnitude is 38 lb.
find magnitude and angle
To find the magnitude and angle of the force vector, we can use the given x and y components.
Given:
Magnitude of x component (Fx): 38 lb
Magnitude of force vector (F): 44 lb
To find the magnitude (F) of the force vector, we can use the Pythagorean theorem:
F = √(Fx^2 + Fy^2)
Since the force vector is in the third quadrant, the y component will be negative. Let's assume the y component as -Fy.
F = √(Fx^2 + (-Fy)^2)
Now, we need to find the value of Fy. Since we know that the magnitude of the y component (Fy) is 44 lb and the magnitude of the x component (Fx) is 38 lb, we can use the equation:
Fy = -√(F^2 - Fx^2)
Substituting the known values, we get:
Fy = -√(44^2 - 38^2)
Fy = -√(1936 - 1444)
Fy = -√(492)
Fy ≈ -22.18 lb (rounded to two decimal places)
Now, let's substitute the values of Fx and Fy into the equation for the magnitude of the force vector:
F = √(Fx^2 + Fy^2)
F = √(38^2 + (-22.18)^2)
F = √(1444 + 491.0324)
F ≈ √(1935.0324)
F ≈ 44 lb (rounded to two decimal places)
So, the magnitude of the force vector is approximately 44 lb.
To find the angle, we can use the tangent function:
Angle (θ) = arctan(Fy/Fx)
θ = arctan((-22.18 lb)/(38 lb))
θ ≈ arctan(-0.5837)
θ ≈ -30.97 degrees (rounded to two decimal places)
Therefore, the magnitude of the force vector is approximately 44 lb and the angle is approximately -30.97 degrees.