Four ropes are tied to a stake, and each is pulled toward a compass direction, N, S, E, or W. A force of 13 N is applied to the rope pulled toward the east. Forces of 23, 25, and 38 N are applied toward the south, west, and north, respectively. What is the direction of the net force exerted on the stake by the ropes?
Fn = (13-25) + i(38-23),
Fn = -12 + i15,
tanAr = Y/X = 15/-12 = -1.25,
Ar = -51.34 deg.
A = 180 + (-51.34) = 128.7 deg.
Well, it seems like this stake is really in a "tug of war" situation! Let's see, 13 N is pulling east, 23 N is pulling south, 25 N is pulling west, and 38 N is pulling north.
So, if we add up all these forces, we get:
13 N (east) + 23 N (south) + 25 N (west) + 38 N (north) = 99 N
Oh boy, that's quite a pull! Now, to find the direction of the net force, we need to look at which direction has the highest overall force. In this case, the 38 N force pulling north is the strongest force, so we can say that the net force is directed toward the north.
So, to answer your question, the direction of the net force exerted on the stake by the ropes is the North direction. It looks like the north has won this little "tug of war" game!
To find the direction of the net force exerted on the stake, we need to calculate the horizontal and vertical components of the forces.
First, let's break down the forces into their horizontal (East-West) and vertical (North-South) components:
- The force of 13 N applied toward the east has an Eastward component of 13 N and a Northward component of 0 N.
- The force of 23 N applied toward the south has a Southward component of 23 N and an Eastward component of 0 N.
- The force of 25 N applied toward the west has a Westward component of 25 N and a Southward component of 0 N.
- The force of 38 N applied toward the north has a Northward component of 38 N and a Westward component of 0 N.
Now let's add up the East-West and North-South components separately:
- East-West component = (13 N Eastward) - (25 N Westward) = -12 N Westward
- North-South component = (23 N Southward) + (38 N Northward) = 61 N Northward
Finally, we can find the direction of the net force by combining both components. Since the East-West component is negative (Westward) and the North-South component is positive (Northward), the net force points Northwest.
Therefore, the direction of the net force exerted on the stake by the ropes is Northwest.
To find the direction of the net force exerted on the stake by the ropes, we need to consider the forces acting in each direction and their magnitudes.
Let's break down the forces along the north-south and east-west axes:
- North-South Forces:
The force applied toward the south is 23 N, while the force applied toward the north is 38 N. Therefore, the net north-south force is 38 N - 23 N = 15 N, acting northward.
- East-West Forces:
The force applied toward the east is 13 N, while the force applied toward the west is 25 N. Therefore, the net east-west force is 13 N - 25 N = -12 N, acting westward.
Now, we have determined the net force values along the north-south and east-west axes. To find the direction of the net force, we need to use the convention of using the angle of force with respect to the positive x-axis.
Using trigonometry, we can calculate the angle of the net force using the following formula:
θ = arctan(y-component of the net force / x-component of the net force)
θ = arctan(15 N / -12 N)
θ ≈ arctan(-1.25) ≈ -51.34°
The negative sign indicates that the net force angle is measured clockwise from the positive x-axis. So, the direction of the net force exerted on the stake by the ropes is approximately 51.34° clockwise from the east, or in other words, it is close to the southeast direction.