Four forces of magnitude 10n 5n 4n and 6n act on an object in the directions north west east and south respectively find the magnitude and direction of their resultant
net N-S ... 10 N - 6 N = 4 N north
net E-W ... 5 N - 4 N = 1 N west
magnitude ... m^2 = 4^2 + 1^2 ... m = √17
direction ... cos(Θ) = 1 / 4
... Θ is the degrees west of north
or ... (90 + Θ) on the unit circle
An object is placed by a force 10N to the east a force of 5N to the north west find the sum of these force given the magnitude and direction the three significant
To find the magnitude and direction of the resultant of these forces, we can use vector addition.
Step 1: Convert each force into its vector form.
Given forces:
- Force 1 (magnitude 10 N) in the north direction: 10N north
- Force 2 (magnitude 5 N) in the west direction: 5N west
- Force 3 (magnitude 4 N) in the east direction: 4N east
- Force 4 (magnitude 6 N) in the south direction: 6N south
Step 2: Draw a vector diagram to depict these forces.
------------> Force 4 (6N)
|
|
|
\/
Force 2 (5N)
Force 1 (10N) ----> Force 3 (4N)
Step 3: Resolve the forces into their horizontal and vertical components.
To resolve the forces into their horizontal (east-west) and vertical (north-south) components, we can use basic trigonometry.
Force 1 (10N):
- Horizontal component: 0 N (It acts in the north direction)
- Vertical component: 10 N (It acts in the north direction)
Force 2 (5N):
- Horizontal component: -5 N (It acts in the west direction)
- Vertical component: 0 N (It does not have any component in the north or south direction)
Force 3 (4N):
- Horizontal component: 4 N (It acts in the east direction)
- Vertical component: 0 N (It does not have any component in the north or south direction)
Force 4 (6N):
- Horizontal component: 0 N (It does not have any component in the east or west direction)
- Vertical component: -6 N (It acts in the south direction)
Step 4: Calculate the resultant horizontal and vertical components.
To calculate the resultant components, we need to add the corresponding components of the forces.
- Horizontal component: 0 N + (-5 N) + 4 N + 0 N = -1 N
- Vertical component: 10 N + 0 N + 0 N + (-6 N) = 4 N
Step 5: Calculate the magnitude and direction of the resultant.
The magnitude of the resultant can be calculated using the Pythagorean theorem:
Resultant magnitude = sqrt((-1 N)^2 + (4 N)^2) = sqrt(1 N + 16 N) = sqrt(17 N) ≈ 4.12 N
The direction of the resultant can be found using trigonometry:
Resultant direction = arctan(Vertical component / Horizontal component)
Resultant direction = arctan(4 N / -1 N) ≈ -75.96 degrees
So, the magnitude of the resultant is approximately 4.12 N, and its direction is approximately 75.96 degrees counterclockwise from the positive x-axis.
To find the magnitude and direction of the resultant force, we need to combine the individual forces using vector addition.
Step 1: Draw a coordinate system. Place the object at the origin of the coordinate system.
Step 2: Draw vectors representing the individual forces. Label each vector with its magnitude and direction.
Step 3: Resolve each force vector into its x-component and y-component. To do this, use trigonometry.
For the force acts in the north direction (10 N), the y-component will be 10 N and the x-component will be 0 N.
For the force acts in the west direction (5 N), the y-component will be 0 N and the x-component will be -5 N (negative because it points in the opposite direction of the positive x-axis).
For the force acts in the east direction (4 N), the y-component will be 0 N and the x-component will be 4 N.
For the force acts in the south direction (6 N), the y-component will be -6 N and the x-component will be 0 N (negative because it points in the opposite direction of the positive y-axis).
Step 4: Add up all the x-components and y-components separately.
Sum of x-components = 0 N - 5 N + 4 N + 0 N = -1 N
Sum of y-components = 10 N + 0 N + 0 N - 6 N = 4 N
Step 5: Use the Pythagorean theorem to find the magnitude of the resultant force.
Magnitude of the resultant force (R) = sqrt((-1 N)^2 + (4 N)^2) = sqrt(1 N^2 + 16 N^2) = sqrt(17 N^2) = 4.12 N (approx.)
Step 6: Use trigonometry to find the direction of the resultant force.
Angle (θ) = arctan(Sum of y-components / Sum of x-components) = arctan(4 N / -1 N) = -75.96°
Since the x-component is negative (-1 N), the angle (-75.96°) is measured clockwise from the positive x-axis.
So, the magnitude of the resultant force is approximately 4.12 N, and its direction is 75.96° clockwise from the positive x-axis.
Well, well, well, looks like we have ourselves a little force party here! Let me grab my clown calculator and crunch some numbers for you.
To find the magnitude of the resultant force, we need to add up all the forces. So, let's do some quick math: 10n + 5n + 4n + 6n = 25n.
Voilà! The magnitude of the resultant force is 25 newtons. That's quite a pull, if you ask me!
Now, for the direction. Since we're dealing with four different forces acting in different directions, we'll need a bit of a compass to guide us.
If we draw a diagram with the forces represented as vectors, we can see that the north and south forces cancel each other out due to their opposite directions. So, we're left with only the west and east forces.
Since the west and east forces have the same magnitude of 10 newtons, we can say that they're balanced, resulting in no net force in the north-south direction. Therefore, the direction of the resultant force is west.
So, in clown terms, the magnitude of the resultant force is 25 newtons, and its direction is west. That's one strong clown pulling in the opposite direction of a circus parade!