Find the resultant in magnitude and direction of forces 10N, 20N, 30N, 40N, acting respectively in the direction 60,120,180 and 270.
I tried drawing out the forces with the respective angles but the angles are more than 360.#confused
nothing wrong with that
you could just use the trig-vector notation
resultant = (10cos60,10sin60) + (20cos120,20sin120) + (30cos180,30sin180) + (40cos270,sin270)
= (5, 5√3) + (-10, 10√3) + (-30, 0) + (0, -40)
= ( -35, 5√3-40 ))
your sketch will confirm that
To find the resultant force, we need to break down each force into its horizontal and vertical components. Then we can add up all the horizontal and vertical components separately to find the net horizontal and vertical forces. Finally, we can find the magnitude and direction of the resultant force using trigonometry.
First, let's calculate the horizontal and vertical components of each force using the given angles.
Force 1 (10N) at an angle of 60 degrees:
Horizontal component = 10N * cos(60) = 5N
Vertical component = 10N * sin(60) = 8.6603N
Force 2 (20N) at an angle of 120 degrees:
Horizontal component = 20N * cos(120) = -10N
Vertical component = 20N * sin(120) = 17.3205N
Force 3 (30N) at an angle of 180 degrees:
Horizontal component = 30N * cos(180) = -30N
Vertical component = 30N * sin(180) = 0N
Force 4 (40N) at an angle of 270 degrees:
Horizontal component = 40N * cos(270) = 0N
Vertical component = 40N * sin(270) = -40N
Now, let's add up all the horizontal and vertical components:
Total horizontal component = 5N - 10N - 30N + 0N = -35N
Total vertical component = 8.6603N + 17.3205N + 0N - 40N = -13.0192N
Using the Pythagorean theorem, we can find the magnitude of the resultant force:
Resultant force = sqrt((-35N)^2 + (-13.0192N)^2) ≈ 37.589N
To find the direction of the resultant force, we can use the inverse tangent function (tan^(-1)):
Direction = tan^(-1)(-13.0192N / -35N) ≈ 20.655 degrees (measured counterclockwise from the positive x-axis)
Therefore, the magnitude of the resultant force is approximately 37.589N and its direction is approximately 20.655 degrees counterclockwise from the positive x-axis.
To find the resultant of the given forces, we first need to resolve each force into its horizontal and vertical components. Since the angles given are more than 360 degrees, we need to convert them to reflex angles by subtracting 360 degrees from each angle.
Let's start with the first force of 10N acting at 60 degrees. The horizontal component of this force can be found using the cosine of the angle, and the vertical component can be found using the sine of the angle.
Horizontal component = 10N * cos(60 - 360) = 5N
Vertical component = 10N * sin(60 - 360) = 8.66N
Next, let's resolve the force of 20N acting at 120 degrees.
Horizontal component = 20N * cos(120 - 360) = -10N (negative because it is in the opposite direction)
Vertical component = 20N * sin(120 - 360) = 17.32N
Similarly, resolve the force of 30N acting at 180 degrees.
Horizontal component = 30N * cos(180 - 360) = -30N
Vertical component = 30N * sin(180 - 360) = -0N
Finally, resolve the force of 40N acting at 270 degrees.
Horizontal component = 40N * cos(270 - 360) = 0N
Vertical component = 40N * sin(270 - 360) = -40N
Now, let's find the resultant horizontal and vertical components by adding up the respective components of all the forces.
Horizontal component = 5N - 10N - 30N + 0N = -35N
Vertical component = 8.66N + 17.32N + 0N - 40N = -13.02N
To find the magnitude of the resultant, use the Pythagorean theorem:
Resultant magnitude = sqrt((-35N)^2 + (-13.02N)^2) = sqrt(1225N^2 + 169.2N^2) = sqrt(1394.2N^2) = 37.32N (approx)
To find the direction of the resultant, use the inverse tangent (arctan) function:
Resultant direction = arctan((-13.02N)/(-35N)) = arctan(0.372) = 21.28 degrees (approx)
Therefore, the resultant of the given forces is approximately 37.32N in magnitude and its direction is 21.28 degrees.