Find the magnitude of the resultant force and the angle between the resultant and each force:
Forces of 2 lbs and 12 lbs at an angle of 60 degrees to each other.
if we let
u = 12.0i
v = 2cos60° + 2sin60° = 1.0i + 1.732j
u+v = 13.0i + 1.732j
in polar form, that is (13.115,7.6°)
To find the magnitude of the resultant force, we can use the Pythagorean theorem:
Magnitude of the resultant force (R) = √(2² + 12²)
= √(4 + 144)
= √148
≈ 12.165 lbs
To find the angle between the resultant and each force, we can use the following trigonometric identities:
1. tanθ = (Perpendicular / Base)
2. θ = atan(Perpendicular / Base)
Let's label the forces as F1 (2 lbs) and F2 (12 lbs).
tan(θ1) = F1 / F2
tan(θ1) = 2 / 12
θ1 = atan(2/12)
θ1 ≈ 9.46 degrees
tan(θ2) = F2 / F1
tan(θ2) = 12 / 2
θ2 = atan(12/2)
θ2 ≈ 78.69 degrees
Therefore, the angle between the resultant and the 2 lbs force is approximately 9.46 degrees, and the angle between the resultant and the 12 lbs force is approximately 78.69 degrees.
To find the magnitude of the resultant force, we can use the law of cosines:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, a = 2 lbs, b = 12 lbs, and C = 60 degrees.
Let's calculate the value of c using the formula:
c^2 = 2^2 + 12^2 - 2(2)(12) * cos(60)
c^2 = 4 + 144 - 48 * cos(60)
To calculate cos(60), we need to convert the angle from degrees to radians:
cos(60) = cos(60 * (pi/180))
cos(60) = cos(pi/3)
cos(60) = 0.5
Substituting this value back into the equation:
c^2 = 148 - 48 * 0.5
c^2 = 148 - 24
c^2 = 124
Now, we take the square root of both sides to find the magnitude of the resultant force:
c = sqrt(124)
c ≈ 11.14 lbs
So, the magnitude of the resultant force is approximately 11.14 lbs.
To find the angle between the resultant and each force, we can use the law of sines:
sin(C) / c = sin(A) / a = sin(B) / b
In this case, C is the angle opposite the resultant force, and A and B are the angles opposite the given forces.
Let's calculate the angle between the resultant and the 2 lbs force (A):
sin(A) / a = sin(C) / c
sin(A) / 2 = sin(60) / 11.14
To calculate sin(60), we need to convert the angle from degrees to radians:
sin(60) = sin(60 * (pi/180))
sin(60) = sin(pi/3)
sin(60) = sqrt(3)/2 ≈ 0.866
Substituting this value back into the equation:
sin(A) / 2 = 0.866 / 11.14
To solve for sin(A), we can rearrange the equation:
sin(A) = (2 * 0.866) / 11.14 ≈ 0.154
Now, we can find the angle A by taking the inverse sine of sin(A):
A = arcsin(sin(A))
A ≈ arcsin(0.154)
A ≈ 8.9 degrees
So, the angle between the resultant and the 2 lbs force is approximately 8.9 degrees.
Similarly, we can calculate the angle between the resultant and the 12 lbs force (B):
sin(B) / b = sin(C) / c
sin(B) / 12 = sin(60) / 11.14
Following the same steps as above, we find:
B ≈ 71.1 degrees
Therefore, the angle between the resultant and the 12 lbs force is approximately 71.1 degrees.
or, if you're not into vectors, try using the law of cosines and law of sines:
the resultant r is
r^2 = 2^2 + 12^2 - 2*2*12 cos120°
r^2 = 172
r = 13.115
the angle θ between the big force and the resultant is
sinθ/2 = sin120°/13.115
sinθ = 0.1321
θ = 7.6°