Three forces of 5 N, 8 N, and 10 N act from the corner of a cube along its edges. Using Cartesian vectors, find the magnitude of the resultant force.
magnitude^2
= hypotenuse^2
= 5^2 + 8^2 + 10^2
= 189
magnitude =√189 = appr 13.75
or
resultant = 5(1,0,0) + 8(0,1,0) + 10(0,0,1)
= (5,8,10)
maginitude of resultant = √(25+64+100) = √189 as above
thank you :)
To find the magnitude of the resultant force, we need to calculate the sum of the individual forces using Cartesian vectors.
First, let's assign Cartesian coordinates to each force. We can consider the three forces as:
- Force A: 5 N in the x-direction (i.e., (5, 0, 0))
- Force B: 8 N in the y-direction (i.e., (0, 8, 0))
- Force C: 10 N in the z-direction (i.e., (0, 0, 10))
Now, we can find the resultant force by adding these vectors. The resultant force, Fresultant, is given by the sum of the individual forces:
Fresultant = Force A + Force B + Force C
Fresultant = (5, 0, 0) + (0, 8, 0) + (0, 0, 10)
Adding the corresponding components, we get:
Fresultant = (5 + 0 + 0, 0 + 8 + 0, 0 + 0 + 10)
Fresultant = (5, 8, 10)
The resultant force vector is (5, 8, 10) in Cartesian coordinates.
To find the magnitude of the resultant force, we can use the Pythagorean theorem in three dimensions:
Magnitude of Fresultant = √(Fresultant_x^2 + Fresultant_y^2 + Fresultant_z^2)
Substituting the values:
Magnitude of Fresultant = √(5^2 + 8^2 + 10^2)
Magnitude of Fresultant = √(25 + 64 + 100)
Magnitude of Fresultant = √189
Therefore, the magnitude of the resultant force is approximately 13.75 N.