Free-Body Diagrams and Newton's Second Law

To construct an accurate FBD, follow these steps:

1. Identify the System: Define the object you are analyzing.
2. Identify Contact Forces: Locate every point where the environment touches the system.
3. Identify Long-Range Forces: Identify forces like gravity that act without physical contact.
4. Draw Vectors: Represent each force as an arrow. The direction of the arrow indicates the direction of the force, and the length of the arrow should be proportional to the force's magnitude.

Common Types of Forces

To draw a correct FBD, you must be able to name and identify the specific forces at play. Here are the most common forces you will encounter:

• Weight ($W$): The force due to gravity. This force always acts vertically downwards.
• Normal Force ($N$): A contact force exerted by a surface. It always acts perpendicular to the surface of contact.
• Tension ($T$): A pulling force transmitted through a rope, wire, or cable. Note that tension is always a pull, never a push.
• Push/Pull: External forces applied directly to the object.
• Friction ($f$): A force that opposes motion. It is categorized into three types:
  • Static friction: Acts on objects that are not yet moving.
  • Dynamic friction: Acts on objects that are currently moving.
  • Rolling friction: Acts on objects with wheels.

From Vectors to Components

Once you have drawn your FBD, you can apply Newton's Second Law to solve for motion. It is often helpful to write the law in two stages: first using vectors, and then breaking the forces down into their components (usually $x$ and $y$ directions).

By breaking forces into components, you can analyze horizontal and vertical motion independently. For example, if an object is being pushed at an angle, you must calculate the $x$-component ($F \cos \theta$) and the $y$-component ($F \sin \theta$) to find the net force in each direction.

Worked Example: The Slowing Elevator

Let's apply these principles to a common problem: A woman in an elevator is moving downwards but slowing down. Draw her FBD.

Step 1: Analyze the Motion

• The velocity is downwards (she is moving down).
• Because she is slowing down, her acceleration must be in the opposite direction of her motion. Therefore, her acceleration is upwards.

Step 2: Determine the Net Force

• According to Newton's Second Law ($\vec{F}_{net} = m\vec{a}$), the direction of the net force is the same as the direction of acceleration.
• Since acceleration is upwards, the net force must be upwards.

Step 3: Draw the FBD

• We have two main forces: Weight ($W$) acting downwards and the Normal Force ($N$) from the elevator floor acting upwards.
• To ensure the net force is upwards, the upward vector ($N$) must be longer than the downward vector ($W$).

Summary Checklist for FBDs:

1. Is the system clearly defined?
2. Are all contact forces (Normal, Tension, Friction, Push) included?
3. Is the weight pointing straight down?
4. Does the relative length of the arrows match the direction of the net force/acceleration?

By following this systematic approach—identifying forces, drawing them as vectors, and applying the component method—you can solve almost any problem in classical dynamics.