Final answer:
To achieve equilibrium, the third ant must apply a force equal in magnitude and opposite in direction to the resultant of the first two ants' forces. We calculate this by breaking down the forces into their components, summing them, and then finding the magnitude and direction of the required force.
Step-by-step explanation:
The question involves finding the necessary force for the third ant to pull to achieve equilibrium. To do this, we'll need to use principles from physics, specifically vector addition and equilibrium.
To find the equilibrium force, the third ant's force must be equal in magnitude and opposite in direction to the resultant force of the first two ants' forces. The first ant pulls with a force of 0.10 N at 45 degrees north of east, and the second ant pulls with 0.25 N at 45 degrees west of north.
We can break these forces down into their x (east-west) and y (north-south) components. For the first ant, the x-component is 0.10 N cos(45) and the y-component is 0.10 N sin(45). For the second ant, the x-component is -0.25 N cos(45) because it's westward, and the y-component is 0.25 N sin(45).
The total x and y components of these two forces are the sum of their individual components. To achieve equilibrium, the third ant's force must cancel out these sums. Hence, its force components must be equal and opposite to the resultant components. We can then use the Pythagorean theorem to find the magnitude of the third ant's required force, which is the square root of the sum of the squares of x and y components. The angle can be found using the inverse tangent function (tan-1) of the y-component divided by the x-component.