Two forces of 3N and 2N are acting at an angle of 45 to each other. Find the magnitude and direction of the resultant force?
3 in x direction
2 /sqrt 2 in x and 2/sqrt2 in y
x force = 3 + 2/sqrt2
y force = 2/sqrt 2
}F} = sqrt [ (3 + 2/sqrt2)^2 + 9 ]
tan angle above x axis = [ (2/sqt2)/(3 +2/sqrt2)]
or
complete the parallelogram
r^2 = 3^2 + 2^2 - 2(3)(2)cos135°
= 13 - 12(-1/√2)
r = appr 4.6 N
or
direction vector = (3,0) + (2cos45, 2sin45)
= (3,0) + (1.414, 1.414) = (4.414 , 1.414)
magnitude = √(4.414^2 + 1.414^2) = appr 4.6
To find the magnitude and direction of the resultant force, we can use the concept of vector addition. We'll break down each force into its x and y components, add them up separately, and then find the magnitude and direction of the resultant force.
First, let's find the x and y components of each force:
Force 1 with magnitude 3N:
Fx1 = 3N * cos(45°)
= 3N * √2/2
= 3√2/2 N
Fy1 = 3N * sin(45°)
= 3N * √2/2
= 3√2/2 N
Force 2 with magnitude 2N:
Fx2 = 2N * cos(45°)
= 2N * √2/2
= √2 N
Fy2 = 2N * sin(45°)
= 2N * √2/2
= √2 N
Now, let's add up the x and y components to find the resultant components:
Rx = Fx1 + Fx2
= 3√2/2 N + √2 N
= (3√2 + 2) / 2 N
Ry = Fy1 + Fy2
= 3√2/2 N + √2 N
= (3√2 + 2) / 2 N
Next, we can find the magnitude (R) and direction (θ) of the resultant force using the Pythagorean theorem and trigonometry:
R = √(Rx^2 + Ry^2)
= √(((3√2 + 2) / 2 N)^2 + ((3√2 + 2) / 2 N)^2)
= √((9/2 + 6√2 + 4/2) N^2 + (9/2 + 6√2 + 4/2) N^2)
= √((9 + 4 + 12√2) N^2)
= √(13 + 12√2) N
θ = atan(Ry / Rx)
= atan(((3√2 + 2) / 2 N) / ((3√2 + 2) / 2 N))
= atan(1)
= 45°
Therefore, the magnitude of the resultant force is √(13 + 12√2) N and the direction is 45°.
To find the magnitude and direction of the resultant force, we can use the concept of vector addition.
Step 1: Resolve the forces into their x and y components.
The force of 3N can be resolved into its x-component and y-component as follows:
- The x-component = 3N * cos(45°)
- The y-component = 3N * sin(45°)
Similarly, the force of 2N can be resolved into its x-component and y-component as:
- The x-component = 2N * cos(45°)
- The y-component = 2N * sin(45°)
Step 2: Add the x-components and y-components separately.
- The x-component of the resultant force = sum of the x-components of the two forces.
= 3N * cos(45°) + 2N * cos(45°)
- The y-component of the resultant force = sum of the y-components of the two forces.
= 3N * sin(45°) + 2N * sin(45°)
Step 3: Find the magnitude of the resultant force using the Pythagorean theorem.
- Magnitude of the resultant force = sqrt( x-component^2 + y-component^2)
Step 4: Find the direction of the resultant force using trigonometry.
- Direction of the resultant force = atan( y-component / x-component)
Now let's calculate it:
x-component of the resultant force = (3N * cos(45°)) + (2N * cos(45°)) = 4.95N
y-component of the resultant force = (3N * sin(45°)) + (2N * sin(45°)) = 4.95N
Magnitude of the resultant force = sqrt((4.95N)^2 + (4.95N)^2) = 6.99N (approx.)
Direction of the resultant force = atan(4.95N / 4.95N) = 45° (approx.)
Therefore, the magnitude of the resultant force is approximately 6.99N, and its direction is 45°.