Two dogs pull horizontally on ropes attached to a post; the angle between the ropes is 66.0∘. Dog A exerts a force of 278N , and dog B exerts a force of 324N .
=Find the magnitude of the resultant force.
=Find the angle the resultant force makes with the rope of dog A.
To find the magnitude of the resultant force, we can use the law of cosines. The formula is:
c^2 = a^2 + b^2 - 2ab * cos(C)
Where:
c is the magnitude of the resultant force,
a is the magnitude of the force exerted by dog A (278N),
b is the magnitude of the force exerted by dog B (324N), and
C is the angle between the ropes (66°).
Substituting the values into the formula, we get:
c^2 = (278N)^2 + (324N)^2 - 2 * 278N * 324N * cos(66°)
Now we calculate:
c^2 = 77284N^2 + 104976N^2 - 180144N^2 * cos(66°)
Next, we substitute the numerical values:
c^2 ≈ 77284 + 104976 - 180144 * 0.40673664307
c^2 ≈ 77284 + 104976 - 73277.555
c^2 ≈ 114982.445
Finally, we take the square root of c^2 to get the magnitude of the resultant force:
c ≈ √(114982.445)
c ≈ 339.26N
Therefore, the magnitude of the resultant force is approximately 339.26N.
To find the angle the resultant force makes with the rope of dog A, we can use the law of sines. The formula is:
sin(A) / a = sin(C) / c
Where:
A is the angle between the resultant force and the force exerted by dog A (the angle we want to find),
a is the magnitude of the force exerted by dog A (278N),
C is the angle between the ropes (66°), and
c is the magnitude of the resultant force (339.26N).
Substituting the values into the formula, we get:
sin(A) / 278N = sin(66°) / 339.26N
Now we calculate:
sin(A) ≈ (278N * sin(66°)) / 339.26N
sin(A) ≈ (278N * 0.91354545764) / 339.26N
sin(A) ≈ 240.47179497 / 339.26N
Finally, we take the inverse sine of sin(A) to get the angle:
A ≈ arcsin(240.47179497 / 339.26N)
A ≈ 41.2°
Therefore, the angle the resultant force makes with the rope of dog A is approximately 41.2°.