Question

Jack pulls east on a rope attached to a camel with a force of 40 ib. Jill pulls north on a rope attached to the same camel with a force of 30 Ib. What is the magnitude and direction of the force on the camel? Assume the vectors lie in a horizontal plane.

Short answer

Answer: The magnitude of the force on the camel is 50 lb, and its direction is 36.9° north of east.

Step-by-step solution

Identify the components of each force

Jack pulls east with a force of 40 lb, and Jill pulls north with a force of 30 lb. Since these vectors lie in a horizontal plane, we don't need to worry about any vertical components. Thus, we can identify the two forces acting on the camel as F1 (Jack's force) and F2 (Jill's force): F1 = 40 lb (east) F2 = 30 lb (north)

Find the net force acting on the camel

To find the net force acting on the camel, we need to sum the forces acting in each direction (east and north) separately: Net force in the east direction (x-axis) = F1 = 40 lb Net force in the north direction (y-axis) = F2 = 30 lb

Calculate the magnitude of the net force

With the net forces in both directions calculated, we can now determine the magnitude of the net force acting on the camel using the Pythagorean theorem. The formula for the magnitude of the net force (F_net) is: F_net = sqrt((net force in the east direction)^2 + (net force in the north direction)^2) Using the values calculated in step 2: F_net = sqrt((40 lb)^2 + (30 lb)^2) = sqrt(1600 + 900) = sqrt(2500) = 50 lb Thus, the magnitude of the net force on the camel is 50 lb.

Find the direction of the net force

To find the direction of the net force, we need to calculate the angle (θ) between the net force and the eastward (x-axis) direction. We can find this angle using the inverse tangent function: θ = arctan(north force / east force) Using the values from step 2: θ = arctan(30 lb / 40 lb) = arctan(0.75) ≈ 36.9° Therefore, the direction of the net force acting on the camel is 36.9° north of the east direction (clockwise from the east axis). The magnitude of the force on the camel is 50 lb, and its direction is 36.9° north of east.

Key concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, particularly used to find relationships between the sides of a right triangle. Simply put, it states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Mathematically, this is written as:

• \( c^2 = a^2 + b^2 \)
• where \( c \) is the hypotenuse, and \( a \) and \( b \) are the triangle's other two sides.

In the context of vector addition, such as the forces Jack and Jill are exerting on the camel, each force vector can be thought of as one side of a right triangle. Here, Jack's pull is along one axis and Jill's pull along another, forming a right angle at their intersection.
This is why the Pythagorean Theorem is employed to find the net or resultant force exerted on the camel, which acts as the hypotenuse of this conceptual triangle. By applying the theorem:

• \( F_{\text{net}} = \sqrt{(40 \text{ lb})^2 + (30 \text{ lb})^2} = 50 \text{ lb} \)

Thus, the theorem helps in calculating the total or net force when two perpendicular force vectors are applied.

Net Force

The concept of net force is crucial when analyzing the effect of multiple forces acting on an object. Net force is essentially the vector sum of all individual forces acting on an object. Here, it tells us how these forces will change the motion of an object.
When multiple forces act upon an object, and they are at right angles to each other, finding the net force can be simplified using vector addition techniques. The forces in our scenario are Jack pulling east with 40 lb and Jill pulling north with 30 lb. Neither of these forces cancels out any component of the other since they are perpendicular.
Following this, the net result is derived from summing these forces using vector addition, resulting in:

• Net east component = 40 lb
• Net north component = 30 lb
• Overall net force is computed using the approach discussed earlier with the Pythagorean Theorem.

Calculating net force by this approach provides both the strength (magnitude) of the force acting on an object as well as its direction.

Trigonometry

Trigonometry deals with the relationships between angles and lengths of triangles, particularly right-angled triangles. It's especially useful for solving problems involving angles and distances.
In vector addition, determining not only the magnitude but also the direction of the resultant vector often requires trigonometric functions. To find the angle of the net force direction, we utilize the inverse tangent function (\( \tan^{-1} \)). This function relates the opposite and adjacent sides of a right triangle:

• \( \theta = \arctan\left(\frac{\text{north force}}{\text{east force}}\right) = \arctan\left(\frac{30}{40}\right) = 36.9^\circ \)

This gives us the angle which the net force makes with the eastward axis, indicating it is 36.9° north of east. Understanding this angle is crucial in navigation and spatial orientation tasks, allowing for accurate prediction and control of movement on the plane.