Two forces of magnitude 8Nand 5N act at an angle 60°to each other. The magnitude of their resultant is
Well, there are two waays:
1. law of cosines
Draw it as a parallelogram with two vectors 60 degrees apart meeting at point A
AD in x direction length 5
AB in direction 60 deg above x axis length 8
Then finish it with BC parallel to AD and DC parallel to AB
we want length AC
well angle BAD = angle BCD = 60
60 + 60 + 2* angle ABC = 360
so angle ABC = 120 degrees
now law of cosines
AC^2 = 8^2 + 5^2 - 2 * 8 * 5 cos 120
AC^2 = 64 + 25 - 80 (-.5) = 64 + 25 + 40 = 129
AC = sqrt (129) = 11.36
Being a physicist though I do it by components
Fx = 5 + 8 cos 60 = 9
Fy = 8 sin 60 = 6.93
|F| = sqrt (81 + 48) = sqrt 129 = 11.36
remarkable same old thing
Fr = 8[60] + 5[0o].
Fr = (8*cos60+5*cos0) + (8*sin60+5*sin0)I,
Fr = 9 + 6.93i = 11.4N[37.6o]
To find the magnitude of the resultant of two forces, you can use the formula:
Resultant = √(F₁² + F₂² + 2F₁F₂cosθ)
Where F₁ and F₂ are the magnitudes of the forces, and θ is the angle between them.
In this case, F₁ = 8N and F₂ = 5N, and the angle θ = 60°. Plugging these values into the formula:
Resultant = √(8² + 5² + 2(8)(5)cos60°)
Resultant = √(64 + 25 + 80cos60°)
Now, we need to find the value of cos60°, which is equal to 0.5.
Resultant = √(64 + 25 + 80(0.5))
Resultant = √(64 + 25 + 40)
Resultant = √(129 + 40)
Resultant = √169
Finally, taking the square root:
Resultant = 13
Therefore, the magnitude of the resultant is 13N.
To find the magnitude of the resultant force, you can use the law of cosines. The formula for the magnitude of the resultant force (R) is:
R² = F₁² + F₂² + 2F₁F₂cosθ
Where:
- F₁ and F₂ are the magnitudes of the two forces (8N and 5N, respectively).
- θ is the angle between the two forces (60°).
Plugging in the values, we have:
R² = (8N)² + (5N)² + 2(8N)(5N)cos(60°)
R² = 64N² + 25N² + 80N²(cos(60°))
R² = 64N² + 25N² + 80N²(0.5)
R² = 64N² + 25N² + 40N²
R² = 129N²
Taking the square root of both sides gives:
R = √129N²
Therefore, the magnitude of the resultant force is √129N.