Several forces act on a particle as shown in the figure below (where F1 = 75.0 N, F2 = 65.0 N, θ1 = 25.0° and θ2 = 74.0°, . If the particle is in translational equilibrium, what are the values of F3 (the magnitude of force 3) and θ3 (the angle that force 3 makes with the x axis)?
F1 is 25 degrees West of North
F2 is 74 degrees South of West
F2 is in the direction South of East
F3 is MINUS the vector sum of F1 and F2. I assume you are familiar with the method of adding vectors by adding the x (east) and y (north) components separately.
To find the values of F3 (the magnitude of force 3) and θ3 (the angle that force 3 makes with the x-axis), we can use vector addition and equilibrium conditions.
1. Start by drawing a rough sketch of the forces F1, F2, and F3, indicating their directions and orientations. Label the angles θ1 and θ2 as given in the problem statement.
2. Now, break down each force into its x and y-components. Since F1 is given as "25 degrees West of North", we can break it down into F1x (horizontal component) and F1y (vertical component). Similarly, for F2, it is given as "74 degrees South of West", so we can break it down into F2x and F2y.
F1x = F1 * sin(θ1)
F1y = F1 * cos(θ1)
F2x = F2 * cos(θ2)
F2y = F2 * sin(θ2)
Substitute the given values to calculate F1x, F1y, F2x, and F2y.
3. Now, let's analyze the equilibrium condition. For translational equilibrium, the net force acting on the particle must be zero in both the x and y directions.
∑Fx = F1x + F2x + F3x = 0
∑Fy = F1y + F2y + F3y = 0
4. Rearrange the above equations to solve for F3x and F3y:
F3x = -F1x - F2x
F3y = -F1y - F2y
5. Determine the magnitude F3 using Pythagorean theorem:
F3 = √(F3x^2 + F3y^2)
6. Calculate the angle θ3 using trigonometry:
θ3 = atan(F3y/F3x), where atan is the inverse tangent function.
7. Substitute the previously calculated values of F3x and F3y into the equation for θ3 and evaluate it.
By following these steps, you should be able to find the values of F3 (the magnitude of force 3) and θ3 (the angle that force 3 makes with the x-axis) when the particle is in translational equilibrium.