Question

Forces with magnitudes of 2000 newtons and 900 newtons act on a machine part at angles of 10° and 85° respectively, with the x-axis. find the direction and the magnitude of the resultant of these forces. round to the nearest tenth place.

Answer 1

To find the magnitude and direction of the resultant force, we can break down each force into its x and y components. Adding up the x-components and y-components, we can use the Pythagorean theorem and inverse tangent function to find the magnitude and direction of the resultant force.

Step-by-step explanation:

To find the magnitude and direction of the resultant force, we can break down each force into its x and y components. The x-component of the 2000 N force is 2000 * cos(10°) = 1997.4 N, and the x-component of the 900 N force is 900 * cos(85°) = 110.32 N. The y-component of the 2000 N force is 2000 * sin(10°) = 347.24 N, and the y-component of the 900 N force is 900 * sin(85°) = 889.88 N.
Adding up the x-components, we get 1997.4 N + 110.32 N = 2107.72 N.
Adding up the y-components, we get 347.24 N + 889.88 N = 1237.12 N.
Using these components, we can find the magnitude of the resultant force using the Pythagorean theorem: √((2107.72 N)^2 + (1237.12 N)^2) ≈ 2424.9 N.
To find the direction of the resultant force, we can use the inverse tangent function: tan^−1(1237.12 N / 2107.72 N) ≈ 30.1°. Therefore, the magnitude of the resultant force is approximately 2424.9 N and the direction is approximately 30.1°.

Answer 2

First, you need to find the components of each force:

F₁x = 2000 cos10 = 1969.6 N
F₁y = 2000 sin10 = 347.3 N
F₂x = 900 cos85 = 78.4 N
F₂y = 900 sin85 = 896.6 N

Then, you have to sum up the same components of the two forces:
Rx = F₁x + F₂x = 1969.6 + 78.4 = 2048.0 N
Ry = F₁y + F₂y = 347.3 + 896.6 = 1243.9 N

In order to find the magnitude of the resultant, you need to apply the Pythagorean theorem:
R = √(Rx² + Ry²)
= √ 2048.0² + 1243.9²
= 2396.2 N

Now, in order to find the direction (angle), you need to use a bit of trigonometry:
α = tan⁻¹ (Ry / Rx)
=tan⁻¹ (1243.9 / 2048)
= tan⁻¹ (0.60737)
=31.3°

Therefore, the answer is: the resultant has a magnitude of 2396.2N with an angle of 31.3° with respect to the x-axys.