# Teaching Forces - Having issues with directions & vectors

• orangeblue
In summary: We can cancel out the common terms and get$$F_{net~both}=(-m_2g)+m_1g=(m_1+m_2)a_{cm}$$which is just the equation for the net force on the system.
orangeblue
Hi everyone. I'm the only physics teacher at my school, so I have nobody to bounce ideas off of. I'm having a problem with students getting confused with direction when calculating net force.
• I teach an introductory physics course. It's the first time the students have ever seen physics and they struggle a LOT at first. We start with Kinematics. To try and simplify things, I ALWAYS call up the positive direction, down negative, right positive, and left negative. The idea is to get them used to assigning a sign to direction. It usually works out fine - students mostly get it.
• In Dynamics, I start to have problems. Here's what I do:
• I define Fnet as the SUM of the forces
• For solving math problems, I've found that using Fnet = (bigger force) - (smaller force) works best for most scenarios. However I run into problems trying to explain why the subtraction sign is there when Fnet is a SUM.
• I tried to tell the students we're making the direction of the bigger force the positive direction, but they just got super confused and I had a hard time explaining it.
Does anyone have a better way of teaching directions/vectors in dynamics? I feel like I'm losing my mind...

I think you should stick to your adopted convention for signs and not worry about the magnitude of Fnet, which is what you're doing. Define Fnet always as the vector sum of all the forces. If, when you do the math, it turns out that Fnet is negative, then it points to the left (or down) and so does the acceleration. It's much cleaner that way.

So I agree that it would be cleaner that way; however, I tend to run into problems when teaching Atwood machines (two masses, connected by a string over a pulley).

The force of gravity on each mass is DOWN, but when we do Fnet we have to make one Fg positive and the other negative for it to work.

How would I explain that to the students? I feel it causes issues when I've been harping so much on calling down negative. I guess I just need to say pulleys change the direction of the force, essentially?

orangeblue said:
So I agree that it would be cleaner that way; however, I tend to run into problems when teaching Atwood machines (two masses, connected by a string over a pulley).

The force of gravity on each mass is DOWN, but when we do Fnet we have to make one Fg positive and the other negative for it to work.

How would I explain that to the students? I feel it causes issues when I've been harping so much on calling down negative. I guess I just need to say pulleys change the direction of the force, essentially?
Yes, an ideal pulley changes the direction of the tension but does not affect its magnitude. Also, note that Fnet is undetermined unless you specify the system on which it acts. To be specific, in the Atwood machine case there are 3 Fnet's possible depending on what you choose as your system
1. Smaller mass: Fnet is positive (up) because the tension (up) is greater than the weight (down). The equation that follows from this is$$F_{net~1}=T-m_1g=m_1a_1$$
2. Larger mass: Fnet is negative (down) because the tension (up) is less than the weight (down). The equation that follows from this is$$F_{net~2}=T-m_2g=m_2a_2$$
Note that the net forces and accelerations are algebraic quantities, they can be positive or negative. However, the magnitudes of the two accelerations are equal, but their directions are opposite which is another way of saying that the string is inextensible. Because the smaller mass is accelerating up and the larger mass down, we write ##a_1=+a## and ##a_2=-a## where ##a## now is a magnitude with the explicit sign showing direction. Hence, the two standard Atwood machine equations found in a standard textbook that provide the standard expression for the acceleration ##a##.

3. Both masses taken together as the system: The mass of the system is ##(m_1+m_2)##. Here ##F_{net~both}## is the net force acting on the center of mass of the two masses. This choice of system can be treated very neatly by "straightening out" the string to vertical (the ideal pulley allows this). The smaller mass is now on top and has force ##m_1g## acting on it conventionally "up"; the larger mass is on the bottom and has force ##-m_2g## acting on it, conventionally "down". Then $$F_{net~both}=(-m_2g)+m_1g=(m_1+m_2)a_{cm}~\rightarrow~a_{cm}=-\frac{m_2-m_1}{m_2+m_1}g.$$

Last edited:
gleem and berkeman

## What are forces and how are they related to directions and vectors?

Forces are interactions between objects that cause changes in their motion. They can be represented by vectors, which have both magnitude (size) and direction. The direction of a force is important because it determines how the object will move in response to the force. For example, a force applied in the forward direction will cause an object to move forward, while a force applied in the opposite direction will cause the object to move backwards.

## How do I determine the direction and magnitude of a force?

In order to determine the direction of a force, you must first identify the two objects that are interacting and the type of force that is acting between them. Then, you can use a diagram or a mathematical formula to represent the force as a vector. The magnitude of a force is typically measured in units of Newtons (N) and can be calculated using the formula F = m x a, where m is the mass of the object and a is the acceleration caused by the force.

## What is the difference between a scalar and a vector quantity?

A scalar quantity is a measurement that has only magnitude, such as mass or temperature. A vector quantity has both magnitude and direction, such as velocity or force. In the context of teaching forces, it is important to understand the difference between scalar and vector quantities because forces are vectors and their direction is crucial in determining how they will affect the motion of an object.

## How can I improve my understanding of directions and vectors in relation to teaching forces?

One way to improve your understanding of directions and vectors is to practice drawing and manipulating them. You can also use real-world examples to help solidify your understanding. Additionally, seeking out additional resources such as textbooks, online tutorials, or consulting with other educators can also be beneficial.

## How can I help students who are struggling with understanding directions and vectors in relation to teaching forces?

One approach to helping students who are struggling is to break down the concept into smaller, more manageable parts. Start with the basics of directions and vectors, and then gradually introduce the concept of forces and their relationship to directions and vectors. Provide hands-on activities or demonstrations to help students visualize and understand the concepts. Additionally, offering extra practice and support, as well as individualized instruction, can also be helpful for struggling students.

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