Free-Body Diagrams

The Forces page introduced many of the forces commonly seen in mechanics: gravity, the normal force, spring-like forces, etc. Although we focused on Newt, a force is not a property of one object. Each force results from an interaction between two things.

An interaction diagram is a way of visualizing the interactions between several objects. Each object gets a bubble, which are then joined by lines that represent the various interactions. The only question is which objects interact, and how.

Note that one interaction link always represents two forces. Tap a link above to see them: one force acts on each object, and the two point in opposite directions. This is Newton’s third law:

The two forces in a pair are equal in size, opposite in direction, and they always act on different objects.

Switch to Newt pushing a box and tap the friction link between Newt and Earth. Newt’s feet push the floor backward, so the floor pushes Newt forward. That forward push from the floor is the only thing that can move Newt as he exerts his own force on the box.

Choosing a System

An interaction diagram attempts to show everything; a free-body diagram shows the forces acting on a particular object or collection of objects. Draw a dashed system boundary around the object or objects you care about.

• A link that crosses the boundary is an external interaction. Its force on the system becomes an arrow on the free-body diagram.
• A link with both ends inside represents two forces that are internal to the system; neither goes on the diagram.

Tap the bubbles below to move them in or out of the system.

Consider the “Newt pushing a box” scenario. With only Newt inside the system, the box pushes him backward. If we add the box to the system, that push vanishes from the diagram. Nothing physical changed; the push just became internal. Choosing a clever system is often the fastest way through a problem, because it hides the forces you do not want to solve for.

For simplicity, the system in a free-body diagram is often drawn as a single dot. Every external force acting on the system is drawn as a vector arrow. When using these to analyze a problem, be sure to only draw:

• Forces. Velocity is not a force. Acceleration is not a force. The quantity $m\vec{a}$ is not a force.
• Forces acting now. The shove that started an object moving is not on the diagram once your hand is gone.

Try each of these on Newt. Choose the forces you think belong, look at the diagram you just described, and then check it.

Newton’s First and Second Laws

Newton’s first law says an object’s velocity does not change unless a net force acts on it. This can be seen in force-free motion when Newt is traveling between walls.

Newton’s second law says the sum of all forces on the system, the net force, sets its acceleration. The Forces page explores how net force and mass trade off; here we apply it to free-body diagrams.

The first law can be considered a special case of the second $\sum \vec{F} = 0$. As in 2D kinematics, each component equation is independent, provided the axes are perpendicular and stay fixed for the whole problem.

Note that this sum is a vector sum. Let’s say Newt stands in an elevator that is speeding up as it rises (accelerating upward). With Newt as the system, there are two forces to consider in the sum:

The magnitude of the normal force has to be greater than Newt’s weight. If the two were equal, the net force would be zero, and Newt would not accelerate. The small leftover arrow in the vector sum diagram represents the net force (be very careful to not think of this as a new force). It points the way Newt is accelerating: up.

Pushing Blocks Together

Push three identical $10\ \text{kg}$ blocks, side by side on a frictionless table, with $96\ \text{N}$. Treating the stack as one system, $a = 96/30 = 3.2\ \text{m/s}^2$. By applying Newton’s second law to individual or pairs of blocks, we can determine that the first block pushes the second with only $64\ \text{N}$, and the second pushes the third with $32\ \text{N}$.

Every kilogram is accelerating at $3.2\ \text{m/s}^2$, so in some sense, each kilogram needs a net force of $3.2\ \text{N}$. Intuitively, one of the contact forces between each pair of blocks must push all the mass ahead of it: $20\ \text{kg}$ is ahead of the first contact, $20 \times 3.2 = 64\ \text{N}$, and $10\ \text{kg}$ is ahead of the second, $10 \times 3.2 = 32\ \text{N}$.

All of this can be reasoned through formally with careful force diagrams and applications of the second law, but how does a block know how hard to push? Again, we turn to a microscopic spring model. Each block is a lattice of atoms held together by spring-like bonds. When the push begins, only the front edge is squeezed. That squeeze travels through the stack as a wave, and eventually settles where every thin slice has just the push from behind needed to keep up with the slice ahead.