Question

Find the resultant of the forces shown in Figure \(5.14\). a. b.

Short answer

Question: Determine the resultant forces for the following cases: a. Three forces acting in the same direction: \(F_1 = 10N\), \(F_2 = 8N\), and \(F_3 = 6N\). b. Two forces acting in opposite directions: \(F_1 = 10N\) and \(F_2 = 7N\). Answer: a. \(R_a = 24N\) b. \(R_b = 3N\)

Step-by-step solution

Case a: Three forces acting in the same direction

For this case, we have three forces acting in the same direction: \(F_1 = 10N\), \(F_2 = 8N\), and \(F_3 = 6N\). To find the resultant of these forces, we simply add them together: \(R_a = F_1 + F_2 + F_3 = 10N + 8N + 6N = 24N\) So the resultant force in case a is \(R_a = 24N\).

Case b: Two forces acting in opposite directions

For this case, we have two forces acting in opposite directions: \(F_1 = 10N\) and \(F_2 = 7N\). To find the resultant of these forces, we subtract the smaller force from the larger one: \(R_b = F_1 - F_2 = 10N - 7N = 3N\) So the resultant force in case b is \(R_b = 3N\).

Key concepts

Force Addition

Understanding force addition is crucial when dealing with multiple forces acting on a single object. Imagine you are pushing a box across the floor with a friend. If both of you push in the same direction, the total force applied to the box is the sum of the individual forces you both exert. Mathematically, if we represent each push as a force vector, force addition is a matter of combining these vectors. For example, if you push the box with a force of 10 Newtons (N), and your friend pushes with 8N, the total force is simply the sum, which amounts to 18N. This summation is directly analogous to adding numerical values:\[ R = F_1 + F_2 + \text{...} + F_n \] where each \( F_i \) is a vector representing a force, and \( R \) is the resultant force vector. If all the forces are acting along the same line and in the same direction, their magnitudes can be directly added.

Vector Sum

Vector summation comes into play when forces are not aligned in the same direction, or when they act at different angles. Unlike scalar quantities, vectors have both magnitude and direction.

Adding Two Vectors

Imagine you're walking your dog, and it's pulling on the leash, trying to chase a squirrel. The dog applies a force in one direction, while you're pulling back in another. The actual path your dog takes depends on the combined effect of these forces. This is a situation where you must consider vector directions. If the forces are represented as arrows, the length of each arrow is proportional to the magnitude of the force, and the arrow points in the direction of the force. To find the resultant force—the vector sum—you would place the tail of the second vector at the head of the first, then draw a vector from the tail of the first vector to the head of the second vector. This new vector represents the combined effect of the two forces.For forces in one dimension, or in the same line, the vector sum simplifies to algebraic addition or subtraction, depending on the directions of the forces. If they are in the same direction, they add positively; if they are in opposite directions, the smaller one subtracts from the larger one.

Opposite Direction Forces

When forces act in opposite directions along the same line, their resultant is found by subtracting the smaller force from the larger force. The direction of the resultant force will be the same as the direction of the larger force.

Effect of Opposite Forces

If you imagine a tug of war between two teams, with one team pulling with a force of 10N to the right and the other with a force of 7N to the left, the resultant force can be found by subtracting the 7N from the 10N, which equals 3N to the right. This opposing force scenario often confuses students, as it involves a subtraction contrary to the intuitive action of addition in the same direction forces.It's also worth noting that if the two opposing forces are equal in magnitude, they cancel each other out, resulting in zero net force—leading to a state of equilibrium. These concepts are frequently observed in statics problems, where the forces in equilibrium keep an object at rest or maintain its constant velocity if it's already in motion.