The three forces shown in the figure below act on a particle (with F1 = 48.0 N, F2 = 58.0 N, θ1 = 63.0°, and θ2 = 27.0°). If the particle is in translational equilibrium, find F3, the magnitude of force 3 and the angle θ3.
F3 = 1 N
θ3 = 2° (counterclockwise from the +x-axis)
Is that your answer? If so, it is wrong. How did you get it? It looks like very a wild guess.
There is no way a 1 N force can balance F1 and F2, which are much larger and in both in the first quadrant. F3 is what is called the equilibrant force. It must be in the third quadrant to balance the other 2 forces.
The way to solve this problem is set the sum of the x components of F1, F2, and F3 equal to zero, and to the same with the y components. That will allow you to solve for F3,x and F3,y.
To find the magnitude of force 3 (F3) and the angle θ3, we need to use the concept of vector addition and equilibrium.
In translational equilibrium, the net force acting on the particle is zero. This means that the vector sum of all the forces acting on the particle is zero.
To find F3 using vector addition, we need to break down each force into its horizontal and vertical components. Let's denote the horizontal component as Fx and the vertical component as Fy.
For force 1 (F1):
Fx1 = F1 * cos(θ1)
Fy1 = F1 * sin(θ1)
Substituting the given values:
Fx1 = 48.0 N * cos(63.0°)
Fy1 = 48.0 N * sin(63.0°)
Similarly, for force 2 (F2):
Fx2 = F2 * cos(θ2)
Fy2 = F2 * sin(θ2)
Substituting the given values:
Fx2 = 58.0 N * cos(27.0°)
Fy2 = 58.0 N * sin(27.0°)
Now, let's find the net horizontal and vertical components of the forces:
Net Fx = Fx1 + Fx2
Net Fy = Fy1 + Fy2
Substituting the values we calculated earlier:
Net Fx = 48.0 N * cos(63.0°) + 58.0 N * cos(27.0°)
Net Fy = 48.0 N * sin(63.0°) + 58.0 N * sin(27.0°)
Since the particle is in translational equilibrium, the net Fx and net Fy must be zero. Therefore:
Net Fx = 0
Net Fy = 0
Solving these equations will give us the magnitude of force 3 (F3) and the angle θ3:
Net Fx = F3 * cos(θ3)
Net Fy = F3 * sin(θ3)
Since F3 = 1 N and θ3 = 2° (counterclockwise from the +x-axis), we can rewrite the equations as:
0 = 1 N * cos(2°)
0 = 1 N * sin(2°)
Solving these equations will give us the answer:
cos(2°) = 0.9994
sin(2°) = 0.0349
Therefore, F3 = 1 N and θ3 = 2°.