Final answer:
Charlie needs to apply a force in the opposite direction of the resultant vector sum of his brothers' forces. By converting the forces into Cartesian vectors based on the bearings, we can calculate the components of Charlie's required force to maintain equilibrium.
Step-by-step explanation:
Step 1: Break down the forces into their x and y components
The force exerted by Noah can be broken down into its x and y components using the following equations:
F_x_Noah = F_Noah × cos(θ_Noah) = 8 N × cos(023°) ≈ 7.13 N
F_y_Noah = F_Noah × sin(θ_Noah) = 8 N × sin(023°) ≈ 3.71 N
The force exerted by Jude can be broken down into its x and y components using the following equations:
F_x_Jude = F_Jude × cos(θ_Jude) = 5 N × cos(155°) ≈ -4.95 N
F_y_Jude = F_Jude × sin(θ_Jude) = 5 N × sin(155°) ≈ -0.58 N
Step 2: Sum the x and y components of the forces
To find the net force in the x-direction, we sum the x-components of the forces exerted by Noah and Jude:
F_x_net = F_x_Noah + F_x_Jude ≈ 7.13 N - 4.95 N ≈ 2.18 N
To find the net force in the y-direction, we sum the y-components of the forces exerted by Noah and Jude:
F_y_net = F_y_Noah + F_y_Jude ≈ 3.71 N - 0.58 N ≈ 3.13 N
Step 3: Calculate the magnitude of the net force
The magnitude of the net force can be calculated using the Pythagorean theorem:
F_net = √(F_x_net^2 + F_y_net^2) ≈ √(2.18 N^2 + 3.13 N^2) ≈ 3.74 N
Step 4: Find the angle of the net force
The angle of the net force can be calculated using the following equation:
θ_net = atan(F_y_net / F_x_net) ≈ atan(3.13 N / 2.18 N) ≈ 55.6°
Step 5: Calculate the force Charlie needs to exert
In order for Charlie to keep the toy fire truck in equilibrium, he needs to exert a force that is equal in magnitude and opposite in direction to the net force. Therefore, the force Charlie needs to exert is 3.74 N at an angle of 55.6° + 180° = 235.6° (since the net force is acting in the third quadrant).