Three forces are applied to the ring of a force table,60N at an angle of 10.0 degrees, 40N at an angle of 70.0 degrees, and 50N at an angle of 100.0 degrees. Find the magnitude of the resultant and the direction by using the analytical method. What is the magnitude of the equilibrant force and its direction?
I am wondering were the angles are measured from? Lets assume measured from the j axis.
break each into components.
60@10=60cos10j+60sin10i
40@70=40cos70j+40sin70i
and so on.
then, add to get the sum. For the equilibrant, change the signs of the sum.
To find the magnitude of the resultant force using the analytical method, we need to break down the given forces into their horizontal and vertical components.
First, let's convert the angles to radians for easier calculations:
Angle 1 (10.0 degrees) = 10.0 * (π/180) = 0.1745 radians
Angle 2 (70.0 degrees) = 70.0 * (π/180) = 1.2217 radians
Angle 3 (100.0 degrees) = 100.0 * (π/180) = 1.7453 radians
Now, we can calculate the horizontal and vertical components for each force:
Force 1:
Horizontal component: 60N * cos(0.1745) = 59.8N
Vertical component: 60N * sin(0.1745) = 10.4N
Force 2:
Horizontal component: 40N * cos(1.2217) = 12.8N
Vertical component: 40N * sin(1.2217) = 35.4N
Force 3:
Horizontal component: 50N * cos(1.7453) = -27.7N (note the negative sign indicating the direction)
Vertical component: 50N * sin(1.7453) = 45.0N
Now, we can find the resultant by summing up the horizontal and vertical components separately:
Horizontal component of the resultant = 59.8N + 12.8N - 27.7N = 44.9N
Vertical component of the resultant = 10.4N + 35.4N + 45.0N = 90.8N
To find the magnitude of the resultant, we can use the Pythagorean theorem:
Magnitude of the resultant = √((Horizontal component)^2 + (Vertical component)^2)
= √((44.9N)^2 + (90.8N)^2)
≈ √(2016.01N^2 + 8256.64N^2)
≈ √10272.65N^2
≈ 101.36N
Therefore, the magnitude of the resultant force is approximately 101.36N.
To find the direction of the resultant, we can use trigonometry:
Direction of the resultant = atan((Vertical component) / (Horizontal component))
≈ atan(90.8N / 44.9N)
≈ atan(2.02)
Using a calculator or reference table, we find that atan(2.02) is approximately 63.25 degrees.
Therefore, the direction of the resultant force is approximately 63.25 degrees.
Now, let's find the magnitude and direction of the equilibrant force.
The equilibrant force is the force needed to balance the resultant force. It has the same magnitude as the resultant force but acts in the opposite direction.
Magnitude of the equilibrant force = 101.36N
Direction of the equilibrant force = 63.25 degrees (opposite direction of the resultant)
Therefore, the magnitude of the equilibrant force is 101.36N, and its direction is 63.25 degrees in the opposite direction of the resultant force.