Given that Force 1 = 70 pounds and Force 2 = 120 pounds , find the angle between the forces if the magnitude of the resultant force is 90 pounds . Round answer to the nearest
degree.
To find the angle between two forces, we can use the law of cosines. The law of cosines states that in any triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of the two sides multiplied by the cosine of the included angle.
In this case, let's call the angle between the two forces θ. We know the magnitudes of the two forces: Force 1 = 70 pounds and Force 2 = 120 pounds. We also know that the magnitude of the resultant force is 90 pounds.
Using the law of cosines, we can relate the magnitudes of the forces and the angle between them to the magnitude of the resultant force:
(90)^2 = (70)^2 + (120)^2 - 2 * 70 * 120 * cos(θ)
Simplifying this equation, we have:
8100 = 4900 + 14400 - 16800 * cos(θ)
Rearranging the equation, we get:
16800 * cos(θ) = 8600
Dividing both sides by 16800, we have:
cos(θ) = 8600 / 16800
Now, we can use the inverse cosine function (cos^-1) to find the angle θ:
θ = cos^-1(8600 / 16800)
Using a calculator, we find that θ is approximately 32.68 degrees. Rounded to the nearest degree, the angle between the forces is 33 degrees.
70^2 + 120^2 + 2*70*120 cosθ = 90^2
cosθ = -2/3
θ = 131.8°
This makes sense, since if the two forces were separated by an acute angle, the resultant would be greater than either of them.
Think of the parallelogram involved.