Problem 34 - College Physics

Question

Two force vectors, \(\overrightarrow{\mathbf{F}}_{1}=(3.0 \mathrm{~N}) \hat{\mathbf{x}}-(4.0 \mathrm{~N}) \hat{\mathbf{y}}\) and \(\overrightarrow{\mathbf{F}}_{2}=(-6.0 \mathrm{~N}) \hat{\mathbf{x}}+(4.5 \mathrm{~N}) \hat{\mathbf{y}},\) are applied to a particle. What third force \(\overrightarrow{\mathbf{F}}_{3}\) would make the net, or resultant, force on the particle zero?

Step-by-Step Solution

Answer

\( \overrightarrow{\mathbf{F}}_3 = 3.0 \mathrm{~N} \hat{\mathbf{x}} - 0.5 \mathrm{~N} \hat{\mathbf{y}} \).

Step 1: Understanding Force Equilibrium

To achieve a resultant force of zero, the sum of all forces acting on the particle must equal zero. This means \( \overrightarrow{\mathbf{F}}_1 + \overrightarrow{\mathbf{F}}_2 + \overrightarrow{\mathbf{F}}_3 = \overrightarrow{0} \). In other words, \( \overrightarrow{\mathbf{F}}_3 \) must be the negative sum of \( \overrightarrow{\mathbf{F}}_1 \) and \( \overrightarrow{\mathbf{F}}_2 \).

Step 2: Calculate the Sum of Forces

Sum the force components along the \(\hat{\mathbf{x}}\) and \(\hat{\mathbf{y}}\) directions separately. For the \(\hat{\mathbf{x}}\) direction: \( 3.0 \mathrm{~N} - 6.0 \mathrm{~N} = -3.0 \mathrm{~N} \). For the \(\hat{\mathbf{y}}\) direction: \(-4.0 \mathrm{~N} + 4.5 \mathrm{~N} = 0.5 \mathrm{~N} \).

Step 3: Determine the Third Force Vector

To make the net force zero, \( \overrightarrow{\mathbf{F}}_3 \) must counterbalance the sum obtained in the previous step. Therefore, \( \overrightarrow{\mathbf{F}}_3 = -(-3.0 \mathrm{~N} \hat{\mathbf{x}} + 0.5 \mathrm{~N} \hat{\mathbf{y}}) = 3.0 \mathrm{~N} \hat{\mathbf{x}} - 0.5 \mathrm{~N} \hat{\mathbf{y}} \).

Key Concepts

Understanding Vector Addition

When dealing with forces as vectors, it's important to remember that each vector has both a magnitude and a direction. In physics, vector addition is a critical tool because it allows us to combine multiple forces to find a resultant vector. This **resultant vector** is simply the vector that results from adding two or more vectors together.

To add vectors, we generally use either graphical methods or components. In this exercise, we're using component addition, which involves summing vectors by their individual coordinates or directions. For instance, a vector like \(\overrightarrow{\mathbf{F}}_1=(3.0 \mathrm{~N})\hat{\mathbf{x}} - (4.0 \mathrm{~N})\hat{\mathbf{y}}\) tells us that there's a positive force of 3.0 N in the x-direction and a negative force of 4.0 N in the y-direction.

• Add vectors in the x and y directions separately: Like this exercise shows: determining the resultant force requires summing the x-components and y-components of the forces separately.


• Keep the direction in mind: Positive or negative signs in vector components indicate direction. In our solution, the negative sign indicates the force is in the opposite direction.

Newton's First Law of Motion

Newton's First Law, also known as the law of inertia, eloquently describes why force equilibrium is essential. It states that an object will remain in its state of motion unless acted upon by a net external force. In simpler terms, if no overall (net) force acts on an object, there will be no change in its velocity—whether that means staying at rest or continuing to move at a constant speed.

In this exercise, you are asked to find a third force \(\overrightarrow{\mathbf{F}}_3\) such that the net force is zero. This is a direct application of Newton's First Law. If \(\overrightarrow{\mathbf{F}}_3\) is correctly calculated, it will create an equilibrium state, where the resultant forces balance each other out, leading to no change in the particle's movement.

• The law applies universally: Whether the object is in motion or at rest, if no net force acts, the state remains unchanged.


• Equilibrium means zero resultant force: Achieving force equilibrium as shown ensures the object retains its state, confirming the principles of Newton's First Law.

What Is Resultant Force?

The resultant force, also known as the net force, is the single force that represents the vector sum of all individual forces acting on an object. It determines the overall effect of these forces. Understanding the resultant force is crucial in the application of physics concepts to real-world problems.

In the exercise, to find this resultant, the vector sum of \(\overrightarrow{\mathbf{F}}_1\) and \(\overrightarrow{\mathbf{F}}_2\) was calculated, and then \(\overrightarrow{\mathbf{F}}_3\) was found such that the overall resultant is zero. This demonstrates how by properly applying forces, one can either start, stop, or maintain an object’s motion.

• Resultant force is the key actor: It determines if and how an object will accelerate.


• It's a calculated vector: Just as shown, by adding up all forces vectorially, you determine this critical factor.