Final answer:
A force of 4.0 kN must be applied to the end of a 2.0 m rod along the x-axis to produce a torque of 8.0 kN·m about the origin.
Step-by-step explanation:
Calculating the Required Force to Produce a Torque on a Rod
To find the force that must be applied to the end of a rod along the x-axis to produce a torque about the origin, we use the formula for torque (τ):
τ = r * F * sin(θ), where r is the distance from the pivot point to the point where the force is applied, F is the force being applied, and θ is the angle between the rod and the force vector.
In this case, the angle θ is 90 degrees since the force is applied perpendicular to the rod.
Since the sin(90°) = 1, the formula simplifies to τ = r * F.
We need a torque of 8.0 kN·m, and the rod is 2.0 m in length (r), so:
τ = r * F
8.0 kN·m = 2.0 m * F
F = 8.0 kN·m / 2.0 m
F = 4.0 kN
The force required to be applied at the end of the rod is 4.0 kN.