To find the magnitude of vector H, we need to first resolve the forces F and G into their vertical and horizontal components. Then, we can use these components to find the net vertical and horizontal forces acting on the particle.
1. Resolving force F:
- The angle of 20' with the vertical means that it makes an angle of 90° - 20' = 89.67° with the horizontal.
- The horizontal component of F, Fx, can be found using cosine: Fx = F * cos(89.67°).
- The vertical component of F, Fy, can be found using sine: Fy = F * sin(89.67°).
2. Resolving force G:
- The angle of 55' with the vertical means that it makes an angle of 90° + 55' = 90.92° with the horizontal.
- The horizontal component of G, Gx, can be found using cosine: Gx = G * cos(90.92°).
- The vertical component of G, Gy, can be found using sine: Gy = G * sin(90.92°).
3. Finding the net horizontal and vertical forces:
- The horizontal component of the net force, F_netx, is the sum of Fx and Gx: F_netx = Fx + Gx.
- The vertical component of the net force, F_nety, is the sum of Fy and Gy: F_nety = Fy + Gy.
4. Finding the magnitude of vector H:
- Using Pythagoras' theorem, the magnitude of H, |H|, can be found: |H| = sqrt(F_netx^2 + F_nety^2).
5. Finding the acute angle with the vertical:
- The acute angle that vector H makes with the vertical, θ, can be found using the inverse tangent: θ = atan(F_nety / F_netx).
Now, let's plug in the values and calculate:
F = 6 N
G = 15 N
Angle F makes with the vertical = 20'
Angle G makes with the vertical = 55'
1. Resolving force F:
Fx = 6 * cos(89.67°) ≈ 0.034 N (horizontal component of F)
Fy = 6 * sin(89.67°) ≈ 5.996 N (vertical component of F)
2. Resolving force G:
Gx = 15 * cos(90.92°) ≈ -0.268 N (horizontal component of G)
Gy = 15 * sin(90.92°) ≈ 14.970 N (vertical component of G)
3. Finding the net horizontal and vertical forces:
F_netx = Fx + Gx ≈ 0.034 N + (-0.268 N) ≈ -0.234 N (horizontal component of the net force)
F_nety = Fy + Gy ≈ 5.996 N + 14.970 N ≈ 20.966 N (vertical component of the net force)
4. Finding the magnitude of vector H:
|H| = sqrt(F_netx^2 + F_nety^2) = sqrt((-0.234 N)^2 + (20.966 N)^2) ≈ 21.021 N