Drawing Free-Body Diagrams in 1D

Introduction

In our last lesson, we focused on the kinematic equations, which are the mathematical tools for describing motion when acceleration is constant. We saw how displacement, velocity, acceleration, and time are all interconnected.

However, kinematics only describes how an object moves; it doesn't explain why. To understand the "why," we need to turn to dynamics and analyze the forces involved. You've already learned about Newton's Laws, which form the bedrock of dynamics. The crucial next step, and the entire focus of today's lesson, is learning how to visually map out all the forces acting on an object.

This map is called a free-body diagram (FBD). Your goal for this lesson is to learn how to construct these diagrams for simple one-dimensional (1D) situations. Mastering this skill is non-negotiable for solving physics problems, as it's the bridge between a real-world scenario and the equations you'll use to analyze it.

What is a Free-Body Diagram?

A free-body diagram is a simplified sketch that isolates an object of interest and represents all the external forces acting on it. It's a way to clear away the clutter and focus only on the physics.

Let's start with a quick video that defines what a free-body diagram is and what its components mean.

To summarize the key conventions:

• The object is simplified, usually represented by a single dot or a box.
• Each force is drawn as an arrow starting from the center of the object, pointing in the direction the force is exerted.
• Each arrow is labeled with the type of force it represents (e.g., $F_{grav}$ for gravity, or $T$ for tension).
• The length of the arrow is proportional to the magnitude (strength) of the force. A stronger force gets a longer arrow.

The Forces: Your Cast of Characters

Before you can draw a diagram, you need to be able to identify which forces are present. For now, we'll focus on a few common ones you'll see in 1D problems.

• Gravitational Force ($F_{grav}$ or $W$): The force of the Earth pulling an object downwards. It's always present for objects near the Earth's surface.
• Normal Force ($F_{norm}$ or $N$): A contact force exerted by a surface to support an object. It acts perpendicular to the surface. For example, a table pushing up on a book.
• Tension Force ($F_{tens}$ or $T$): The pulling force transmitted through a string, rope, cable, or wire. It always pulls away from the object.
• Friction Force ($F_{frict}$ or $f$): A contact force that opposes the motion or attempted motion between surfaces.
• Applied Force ($F_{app}$): A general term for a force exerted on an object by a person or another object, like a push or a pull.
• Air Resistance ($F_{air}$): A type of friction force that acts on objects as they move through the air. It opposes the motion.

How to Draw a Free-Body Diagram: A Step-by-Step Method

Drawing an FBD is a methodical process. Following these steps consistently will help you avoid mistakes.

Here is a summary of that process:

1. Isolate the Object: Mentally (or on paper), separate the single object you are analyzing from its environment (the "free body").
2. Identify Forces: Go through your list of force types. Ask yourself:
   • Is the Earth pulling on the object? (Gravity)
   • Is the object touching any surfaces? (Normal force, Friction)
   • Are there any ropes, strings, or cables attached? (Tension)
   • Is something actively pushing or pulling it? (Applied Force)
   • Is it moving through the air? (Air Resistance)
3. Draw and Label: Draw a box or dot for the object. For each force you identified, draw a labeled arrow starting from the center of the box, pointing in the correct direction.
4. Consider Arrow Lengths: This is where you connect back to Newton's First and Second Laws.
   • If the object is at rest or moving with constant velocity, the net force is zero. This means forces in opposite directions must be equal in magnitude. Draw the opposing arrows with the same length.
   • If the object is accelerating, there is a net force. The force in the direction of acceleration must be larger than any opposing force. Draw the arrow in the direction of acceleration longer than the one pointing opposite.

Guided Examples

Let's see this process in action with a few 1D scenarios.

Let's break down two key cases from that video:

1. Book at rest on a table:

   • Forces: Gravity pulls down, the table pushes up with a normal force.
   • Motion: At rest, so acceleration is zero.
   • Conclusion: The forces are balanced. The upward normal force arrow must be the same length as the downward gravity arrow.

2. Skydiver speeding up while falling:

   • Forces: Gravity pulls down, air resistance pushes up.
   • Motion: Accelerating downwards ("speeding up").
   • Conclusion: There is a net downward force. The downward gravity arrow must be longer than the upward air resistance arrow.

A Deeper Dive: The Normal Force vs. Action-Reaction

You've expressed an interest in understanding physics from "first principles," so let's address a very common point of confusion. In the 'book on a table' example, you see an upward normal force and a downward gravitational force that are equal and opposite. It's tempting to call these an action-reaction pair from Newton's Third Law, but they are not.

This distinction is critical for correctly drawing FBDs. Remember, an FBD shows forces acting on one object. Action-reaction pairs always act on two different objects.

This next video gives an excellent, detailed explanation of this concept.

The key takeaway is that your free-body diagram for the book should only include the forces acting on the book: the Earth pulling the book down (gravity) and the table pushing the book up (normal force).

Practice Your Skills

The only way to get comfortable with FBDs is to draw them. The following resource provides several practice scenarios. Try to draw the diagram for each situation first on your own, then check your work against the provided answer.

Connection to Robotics

Your interest in robotics provides a perfect context for FBDs. Imagine programming a simple robotic gripper to pick up a block.

If we draw an FBD for the block being held stationary by the gripper:

• Gravity ($W$) pulls the block down.
• The gripper's "fingers" squeeze the block. This squeezing provides an inward Normal Force ($N$) from each side.
• This normal force creates an upward Static Friction force ($f_s$) between the fingers and the block.

For the robot to successfully hold the block without it slipping, the FBD tells us that the upward friction force must be equal to (or greater than) the downward force of gravity ($f_s\ge W$). As a programmer, you would use this principle to calculate the minimum squeezing force ($N$) the motors need to apply to generate enough friction.

Conclusion

Today, you've learned the fundamental skill of translating a physical situation into a free-body diagram. This is the first and most important step in solving almost any problem involving forces.

Key Takeaways:

• A free-body diagram isolates a single object and shows all the external forces acting upon it.
• The state of motion (at rest, constant velocity, or accelerating) determines the relative lengths of the force vectors. If acceleration is zero, opposing forces are balanced.
• It's crucial to only include forces acting on the object of interest. Forces like weight and normal force acting on the same resting object are not an action-reaction pair.

Preview of the Next Lesson:

Now that you can confidently draw the forces on an object, we're ready to put them to use. In our next lesson, we will apply Newton's Second Law ($\Sigma F=ma$) to the free-body diagrams you've learned to draw. You'll see how to turn your diagrams into mathematical equations to solve for unknown quantities like acceleration or force in 1D problems.