Final answer:
The force exerted on the crate at the final position can be calculated using trigonometry and the properties of a right triangle to relate the tension in the rope, the weight of the crate, and the angle created by the crate's displacement.
Step-by-step explanation:
The magnitude of the force F exerted on a 230 kg crate to move it a distance d of 4 m to the side, when the rope from which the crate hangs has a length L of 12 m, can be found by considering the equilibrium condition of the system in its final position. The crate will form a right triangle with the rope and the horizontal distance it has moved. The force F will be equal to the horizontal component of the tension in the rope. To find the tension, we can use trigonometry and the fact that the vertical component of the tension must balance the weight of the crate (mg, where m is the mass and g is the acceleration due to gravity).
The tension T in the rope is given by T = mg / cos(θ), where cos(θ) = L / hypotenuse and the hypotenuse can be found using the Pythagorean theorem (hypotenuse = √(L^2 + d^2)). Once we have the tension, we can determine the horizontal force F by multiplying the tension by sin(θ), since sin(θ) = d / hypotenuse. This gives F = T * sin(θ) and substituting the earlier equation for T gives F = mg * sin(θ) / cos(θ). Simplifying, we find F = mg * tan(θ), where tan(θ) = d / L.