Two forces of 7 lb. and 14 lb. act on a body at right angles to each other. Find their resultant. Find the angle their resultant force makes with the force of 14 lb.
To find the resultant of two forces, we can use the Pythagorean theorem. The theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs.
Let's consider the 7 lb. and 14 lb. forces as the two legs of a right triangle. The 7 lb. force acts along one leg, and the 14 lb. force acts along the other leg. We need to find the length of the hypotenuse, which represents the resultant force.
Using the Pythagorean theorem, we can calculate the magnitude of the resultant force:
Resultant force^2 = (7 lb.)^2 + (14 lb.)^2
Resultant force^2 = 49 lb^2 + 196 lb^2
Resultant force^2 = 245 lb^2
Taking the square root of both sides, we get:
Resultant force = √(245 lb^2) ≈ 15.68 lb.
Therefore, the magnitude of the resultant force is approximately 15.68 lb.
To find the angle that the resultant force makes with the 14 lb. force, we can use trigonometry. We will consider the 14 lb. force as the adjacent side and the resultant force as the hypotenuse. Let's call the angle between the resultant force and the 14 lb. force "θ."
Using the cosine function:
cos(θ) = adjacent / hypotenuse
cos(θ) = 14 lb. / 15.68 lb.
Solving for θ, we take the inverse cosine (or arccosine) of both sides:
θ = cos^(-1)(14 lb. / 15.68 lb.)
Calculating:
θ ≈ 28.07 degrees.
Therefore, the angle that the resultant force makes with the 14 lb. force is approximately 28.07 degrees.