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C. Forces · Forces

Free-Body Diagrams and Net Force

C2.1 C2.2 A1.12About 70 minutes
Learning goals
  • I can draw a free-body diagram showing every force acting on one object.
  • I can choose a positive direction and add forces along a line to find the net force.
  • I can tell from the net force whether an object's velocity is changing.
  • I can use balanced forces to find an unknown force.

Words to know

system diagram
A simple sketch of a situation showing all the objects involved.
free-body diagram (FBD)
A diagram of a single object, drawn as a box or dot, with an arrow for every force acting on it. Each arrow starts at the object and points in the force's direction.
net force (Fnet)
The vector sum of all the forces acting on an object. Also called the resultant or unbalanced force. Written ΣF in some books.
balanced forces
Forces whose vector sum is zero, so Fnet = 0 N.
sign convention
The choice of which direction is positive (for example right and up are +). Forces the other way are negative.

One object, many forces

Picture a toboggan being pulled up a snowy trail. Gravity pulls it down. The snow pushes it up. A rope pulls it forward. Friction drags it back. What will the toboggan actually do?

To answer, physicists use one of the most useful tools in all of physics: the free-body diagram. It strips away everything except the object and the forces acting on it, so you can add them up. The sum is the net force, and the net force is what decides how the velocity changes.

How to draw a free-body diagram

  1. Pick one object. Draw it as a box or a dot. (Only forces acting on this object go in the diagram, never forces it exerts on other things.)
  2. Gravity first. Draw Fg straight down from the object.
  3. Go around the edge. For everything that touches the object, draw the contact force it gives: FN perpendicular to each surface, Ff along the surface opposing sliding, FT along each rope, Fa for each push.
  4. Add any other non-contact forces (magnetic, electric), if there are any.
  5. Label every arrow with its symbol and, if known, its size. Make lengths roughly match the sizes.
  6. Show the positive direction with a small arrow in the corner, e.g. "+ right, + up".

Do not draw velocity or acceleration as force arrows. "The force of motion" is not a force.

boxSystem diagram (a sketch)Free-body diagramboxFNFgFTFf
Left: a sketch of the situation. Right: the free-body diagram shows only the box and the four forces acting on it. The person, the rope and the floor are replaced by the forces they exert.

Finding the net force in one dimension

In this unit all motion is along a line, so we treat horizontal and vertical forces separately.

  • Choose a positive direction (usually the direction of motion, or right and up).
  • Forces that way are +, forces the opposite way are −.
  • Add them: Fnet = F₁ + F₂ + F₃ + ...

A positive answer means the net force points the positive way; a negative answer means it points the other way.

If the object is not speeding up or slowing down vertically (say it stays on the floor), the vertical forces must balance: the vertical net force is zero. That is how we usually find the normal force on a level surface: FN = Fg, as long as nothing else pushes up or down.

Worked example

Example 1: A box dragged across the floor

A 2.0 kg box is pulled to the right across a level floor by a horizontal rope with a tension of 15 N. Kinetic friction on the box is 6.0 N. Draw the FBD and find the net force.

  1. Forces: gravity down, normal force up, tension right, friction left (opposing the sliding).
  2. Gravity: Fg = mg = (2.0 kg)(9.8 N/kg) = 19.6 N down.
  3. Vertical: the box stays on the floor, so the vertical forces balance: FN = 19.6 N up. Vertical net force = 19.6 − 19.6 = 0.
  4. Horizontal, with right as +: Fnet = +15 N + (−6.0 N) = +9.0 N.
  5. The net force is 9.0 N to the right, the same direction as the motion, so the box will speed up.

Fnet = 9.0 N right (the vertical forces cancel).

Worked example

Example 2: Tug of war

In a tug of war at a winter carnival, team A pulls left with 1250 N and team B pulls right with 1180 N. Find the net force on the rope.

  1. Choose right as positive.
  2. Fnet = +1180 N + (−1250 N) = −70 N.
  3. The negative sign means the net force points left.

Fnet = 70 N left, toward team A.

Adding forces along a line (right = +)+1180 N (team B)−1250 N (team A)Fnet = −70 N (70 N left)(net force: from first tail to last tip)
Forces along one line add like signed numbers. Place the arrows tip to tail: +1180 N + (−1250 N) = −70 N. The short arrow is the net force, 70 N to the left.
Worked example

Example 3: Finding an unknown force

A sled is pulled forward with an applied force of 85 N. The net force on it is 30 N forward. What is the friction force?

  1. Forward is +. The only horizontal forces are Fa = +85 N and friction Ff (backward, unknown size).
  2. Fnet = Fa + Ff, so +30 N = +85 N + Ff.
  3. Ff = 30 − 85 = −55 N.

Friction is 55 N backward.

Common mistakes

  • Drawing forces the object exerts (the box pushing down on the floor). Only forces on the object belong in its FBD.
  • Forgetting the sign: always write the + direction before adding.
  • Assuming the net force must point the way the object moves. A sliding puck moves right while the net force (friction) points left.
Just for fun

Physics teachers call it a free-body diagram. Students sometimes call it "a box having a bad hair day": arrows sticking out in every direction.

Hands-on activity

Bathroom-scale free-body diagrams

You need: a bathroom scale, a sturdy kitchen counter or table, paper and a pencil.

A bathroom scale measures the normal force it pushes up on you with.

  1. Stand still on the scale. Record the reading. Draw your FBD: Fg down, FN (from the scale) up. They balance.
  2. Stand on the scale beside the counter and press down gently on the counter with your hands. What happens to the reading? Draw the FBD: the counter now pushes up on your hands too, so the scale has to push up less.
  3. Hold a heavy book or a bag of potatoes while standing on the scale. Predict, then check the new reading. Draw the FBD of you (the book pushes down on your hands) and of the book.
  4. (Only if your scale shows numbers quickly.) Bend your knees quickly downward, then push up. Watch the reading jump around. You will explain this in the lesson on mass and weight.

Stay steady: hold the counter edge and do not stand on anything wobbly.

Your turn

1

Two forces act on a crate: 40 N right and 25 N left. Find the net force.

2

Three forces act on a puck: 12 N east, 7 N east and 20 N west. Find the net force.

3

A 5.0 kg bag of potatoes hangs at rest from a rope. Draw its FBD and find the tension.

4

A car's driving force from the road is 3200 N forward. Air resistance is 900 N and rolling friction is 400 N, both backward. Find the net force.

5

A model rocket has a thrust of 12 N up. Its weight is 2.5 N and air resistance is 1.5 N. Find the net force while it rises.

6

Draw an FBD for a person standing in an elevator that is moving up at a constant speed. How do the forces compare?

More practice: warm-up

1

What does a free-body diagram show?

2

If the forces on an object are balanced, what is the net force?

3

Forces of 30 N up and 30 N down act on a lamp. Net force?

4

Forces of 8 N left and 3 N left act on a block. Net force?

5

Why do we leave the force a box exerts on the floor out of the box's FBD?

More practice: on your own

1

A box is pushed right with 60 N. The net force on it is 18 N right. Find the friction force.

2

A 3.0 kg book lies on a table, and your hand pushes straight down on it with 10 N. Find the normal force from the table.

3

A 0.50 kg lamp hangs at rest from two vertical strings that share the load equally. Find the tension in each.

4

A 70 kg skydiver feels 500 N of air resistance. Find the net force on her.

5

In a tug of war each team pulls with 800 N. What is the net force on the rope, and what can you say about its motion?

Challenge

1

A 4.0 kg box rests on the floor. You pull straight up on it with a rope, with 15 N, but it does not lift. Find the normal force. What pull would make the normal force zero?

2

Two tugboats each pull a barge forward with 2.5 × 10⁴ N. Water resistance on the barge is 3.8 × 10⁴ N. Find the net force.

3

A 2.0 kg book sits on top of a 3.0 kg book on a table. Draw the FBD of the bottom book and find each force.

Check your understanding

Quiz

1.

Which force does not belong in the free-body diagram of a book resting on a table?

2.

Forces of 25 N right, 10 N left and 5 N left act on a cart. The net force is:

3.

A crate slides to the right but the net force on it points left. The crate is:

4.

A 6.0 kg backpack hangs at rest on a hook. The force from the hook is about:

5.

If Fnet = 0 on an object, then its velocity is:

6.

A sled is pulled with 50 N forward and the net force is 20 N forward. Friction is:

For parents and teachers

Big idea: Free-body diagrams (C2.2, and the representation skill A1.12) are the single most important tool in this unit. Every later lesson starts with one.

How to help: When your teen solves a problem, ask to see the FBD first. Check: one object only; gravity drawn; one contact force for each thing touching; every arrow labelled; a + direction shown.

Common difficulties:

  • Including forces the object exerts on other things.
  • Drawing a "motion" arrow as if it were a force.
  • Forgetting negative signs when adding.
  • Assuming the normal force always equals the weight. It does only when nothing else pushes or pulls vertically and there is no vertical acceleration (see the challenge questions).