During a picnic, you and two of your friends decide to have a three-way tug-of-war, with three ropes in the middle tied into a knot. Roberta pulls to the west with 412 N of force; Michael pulls to the south with 580 N. In what direction and with what magnitude of force should you pull to keep the knot from moving? (Let the eastward direction be the the +x direction.)

Question

During a picnic, you and two of your friends decide to have a three-way tug-of-war, with three ropes in the middle tied into a knot. Roberta pulls to the west with 412 N of force; Michael pulls to the south with 580 N. In what direction and with what magnitude of force should you pull to keep the knot from moving? (Let the eastward direction be the the +x direction.)

magnitude 1
direction 2 ° (counter-clockwise from the +x-axis)

Answer 1

To find the direction and magnitude of the force that you should pull to keep the knot from moving, we can use vector addition.

First, let's represent the forces exerted by Roberta and Michael as vectors. The force exerted by Roberta is 412 N to the west, so we can represent it as a vector:

Roberta's force (R): -412 N

Similarly, the force exerted by Michael is 580 N to the south, so we can represent it as a vector:

Michael's force (M): -580 N

Now, let's find the resulting force of Roberta's and Michael's forces. We can do this by adding the two vectors:

Resultant force (F) = R + M

To add the vectors, we need to break them down into their x and y components. The x-axis is east-west, and the y-axis is north-south.

Roberta's force can be broken down as follows:
R = (-412 N, 0 N)

Michael's force can be broken down as follows:
M = (0 N, -580 N)

Now, let's add the x and y components separately to get the resultant force:

Fx = Rx + Mx = -412 N + 0 N = -412 N
Fy = Ry + My = 0 N + (-580 N) = -580 N

So, the resultant force F is (-412 N, -580 N).

To find the magnitude of the force (magnitude 1), we can use the Pythagorean theorem:

|F| = sqrt(Fx^2 + Fy^2)

|F| = sqrt((-412 N)^2 + (-580 N)^2)
= sqrt(169744 N^2 + 336400 N^2)
= sqrt(506144 N^2)
= 711 N (approx.)

The magnitude of the force you should pull is approximately 711 N.

To find the direction of the force (direction 2 ° counter-clockwise from the +x-axis), we can use trigonometry. The angle can be found using the inverse tangent (arctan) function:

θ = arctan(Fy/Fx)

θ = arctan((-580 N)/(-412 N))
= arctan(1.4087)
= 52.9 ° (approx.)

Therefore, the direction of the force you should pull is approximately 52.9 ° counter-clockwise from the +x-axis.

So, to keep the knot from moving, you should pull with a magnitude of approximately 711 N in a direction approximately 52.9 ° counter-clockwise from the +x-axis.