Three forces with magnitudes of 70 pounds, 40 pounds, and 60 pounds act on an object at angles of -30 degrees, 45 degrees, and 135 degrees, respectively, with the positive x-axis. Find the direction and manitude of the resultant of these forces.
Please explain. I really do not understand these.
The three vectors can be written as
(70cos(-30),70sin(-30)) or (60.622, -35)
(40cos45,40sin45), or (28.2843, 28.2843)
(60cos135,60sin135) or -42.4264, 42.4264)
so the sum of those three vectors is
(46.4799, 35.7107)
the magnitude of that is
√(46.4799^2 + 35.7107^2) = 58.614
the direction would be arctan(35.7107/46.4799) = 37.535°
Thank You!
THE SAME AS HANNAH!
Well, it seems like those forces are really pulling in different directions. It's like they can't make up their minds on where to go. But don't worry, I'll help you figure it out!
To find the resultant force, we need to add up all the forces acting on the object. But since they’re at different angles, we first need to break them down into their components.
Let's start with the force of 70 pounds at an angle of -30 degrees. To break it down into its x and y components, we'll use some trigonometry. The x-component can be found by multiplying the force by the cosine of the angle, so the x-component of this force is 70 * cos(-30). Similarly, the y-component can be found by multiplying the force by the sine of the angle, so the y-component is 70 * sin(-30).
Now let's move on to the force of 40 pounds at an angle of 45 degrees. Again, we use trigonometry to break it down. The x-component is 40 * cos(45), and the y-component is 40 * sin(45).
Lastly, we have the force of 60 pounds at an angle of 135 degrees. We apply the same trigonometry once again. The x-component is 60 * cos(135), and the y-component is 60 * sin(135).
Now that we have all the x and y components of the forces, we can add them up to find the resultant. We add up all the x-components together, and all the y-components together. The magnitude of the resultant force can be found using the Pythagorean theorem, which states that the magnitude of the resultant force squared is equal to the sum of the squares of its x and y components. Finally, the direction can be found by taking the inverse tangent of the y-component divided by the x-component.
So go ahead and do those calculations, and remember to keep your sense of humor intact while you're at it! Good luck!
To find the resultant of these forces, we need to break down each force into its horizontal (x-component) and vertical (y-component) vectors. We can then add the x-components together and the y-components together to find the resultant components. Finally, we can use the Pythagorean theorem to find the magnitude of the resultant and trigonometry to find its direction.
Step 1: Resolve the forces into x and y components:
To find the x-component of a force, we multiply the force magnitude by the cosine of the angle it makes with the positive x-axis. Similarly, to find the y-component, we multiply the force magnitude by the sine of the angle.
Force 1:
Magnitude = 70 pounds
Angle = -30 degrees
x-component of Force 1 = 70 * cos(-30°)
y-component of Force 1 = 70 * sin(-30°)
Force 2:
Magnitude = 40 pounds
Angle = 45 degrees
x-component of Force 2 = 40 * cos(45°)
y-component of Force 2 = 40 * sin(45°)
Force 3:
Magnitude = 60 pounds
Angle = 135 degrees
x-component of Force 3 = 60 * cos(135°)
y-component of Force 3 = 60 * sin(135°)
Step 2: Find the sum of the x-components and y-components:
Add up all the x-components and y-components separately:
Sum of x-components = x-component of Force 1 + x-component of Force 2 + x-component of Force 3
Sum of y-components = y-component of Force 1 + y-component of Force 2 + y-component of Force 3
Step 3: Find the magnitude and direction of the resultant:
The magnitude of the resultant force can be found using the Pythagorean theorem:
Resultant magnitude = sqrt((Sum of x-components)^2 + (Sum of y-components)^2)
The direction of the resultant force can be found using:
Resultant direction = atan(Resultant y-component / Resultant x-component)
Now, let's calculate each component and then find the resultant magnitude and direction.
x-component of Force 1 = 70 * cos(-30°) = 70 * √(3)/2 = 60.62 pounds
y-component of Force 1 = 70 * sin(-30°) = -70 * 1/2 = -35 pounds
x-component of Force 2 = 40 * cos(45°) = 40 * √(2)/2 = 28.28 pounds
y-component of Force 2 = 40 * sin(45°) = 40 * √(2)/2 = 28.28 pounds
x-component of Force 3 = 60 * cos(135°) = 60 * -√(2)/2 = -42.43 pounds
y-component of Force 3 = 60 * sin(135°) = 60 * √(2)/2 = 42.43 pounds
Sum of x-components = 60.62 + 28.28 - 42.43 = 46.47 pounds
Sum of y-components = -35 + 28.28 + 42.43 = 35.71 pounds
Resultant magnitude = sqrt((46.47)^2 + (35.71)^2) = 59.51 pounds
Resultant direction = atan(35.71 / 46.47) = 41.63 degrees
Therefore, the resultant of these forces is approximately 59.51 pounds in magnitude, and it makes an angle of 41.63 degrees with the positive x-axis.