A force vector points at an angle of 78.2 ° above the +x axis. It has a y component of +190 newtons (N). Find (a) the magnitude and (b) the x component of the force vector.
If the vector has magnitude v, then
v sin78.2° = 190
and then x = v cos78.2° (or, x^2 + 190^2 = v^2)
To find the magnitude and the x-component of the force vector, we can use trigonometry.
(a) To find the magnitude of the force vector, we can use the Pythagorean theorem, which states that the square of the magnitude (hypotenuse) of a right triangle is equal to the sum of the squares of the other two sides. In this case, the magnitude of the force vector is the hypotenuse of the right triangle formed by the x-component and the y-component of the force vector.
Using trigonometry, we can relate the y-component and the magnitude of the force vector using the sine function:
sin(θ) = y-component / magnitude
Rearranging the formula and solving for the magnitude, we get:
magnitude = y-component / sin(θ)
In this case, the y-component is +190 N and the angle is 78.2°. Plugging these values into the formula, we get:
magnitude = 190 N / sin(78.2°)
Using a calculator, we can evaluate sin(78.2°) to be approximately 0.97845. Substituting this value, we get:
magnitude ≈ 190 N / 0.97845 ≈ 194.13 N
Therefore, the magnitude of the force vector is approximately 194.13 N.
(b) To find the x-component of the force vector, we can use trigonometry again, but this time, we will use the cosine function:
cos(θ) = x-component / magnitude
Rearranging the formula and solving for the x-component, we get:
x-component = magnitude * cos(θ)
In this case, the magnitude is 194.13 N and the angle is 78.2°. Plugging these values into the formula, we get:
x-component = 194.13 N * cos(78.2°)
Using a calculator, we can evaluate cos(78.2°) to be approximately 0.20618. Substituting this value, we get:
x-component ≈ 194.13 N * 0.20618 ≈ 40.01 N
Therefore, the x-component of the force vector is approximately 40.01 N.