Aim for FivePhysics C: Mechanics

Unit 2 · Topic 2.3

2.3 Newton’s Third Law

Newton's third law says forces come in pairs: if A pushes on B, B pushes back on A with a force of the same size in the opposite direction. The two forces act on different objects, so they never cancel on one free-body diagram. Internal force pairs also can't change the motion of a system's center of mass.

Key terms

  • Newton's third law
  • force pair
  • internal force
  • tension
  • ideal string
  • ideal pulley

Force pairs

Every force is half of an interaction. When object A exerts a force on object B, B exerts a force on A that is:

  • equal in size,
  • opposite in direction,
  • the same type of force (both gravitational, both normal, both friction…), and
  • acting on the other object.

Spotting a real third-law pair

Name each force as "the force of X on Y". Its partner is "the force of Y on X". For a book resting on a table:

The pair of Earth's gravity on the book is the book's gravity pulling Earth up. The pair of the table's normal force on the book is the book pushing down on the table.

The book's weight and the normal force on it are equal and opposite here, but they are not a third-law pair. They act on the same object and are different types. They're equal only because the book isn't accelerating, which is Newton's second law, not the third. In an accelerating elevator they wouldn't be equal, but each third-law pair still would be.

The sizes in a pair are equal even when the objects are very different. When a truck hits a small car, the force of the truck on the car equals the force of the car on the truck. The car's acceleration is bigger because its mass is smaller.

Internal forces and the center of mass

If both objects in a pair are inside your system, the two forces are internal and add to zero. So internal forces can't change the velocity of the system's center of mass. You can't lift yourself by pulling on your own belt.

This is why choosing a system matters. For two blocks pushed together, the contact forces between them are internal if the system is both blocks, and they drop out of the equation for the system's acceleration.

Ideal strings and pulleys

An ideal string (or rope) is massless and doesn't stretch. Because it has no mass, the net force on any piece of it must be zero, so the tension is the same all along it. It pulls on the objects at both ends with that same tension. A rope with real mass is different: the tension can change along it, because lower parts of a hanging rope must also hold up the rope's own weight.

An ideal pulley is massless and frictionless. It changes the direction of a string's pull without changing the tension. Later, with rotation (Unit 5), a pulley with mass can have different tensions on its two sides.

Because an ideal string doesn't stretch, objects tied together by it move with the same speed and the same size of acceleration.

Worked examples

Try each one yourself first, then open the solution.

  1. Example 1Calculator allowed

    Pushing two blocks

    Two blocks sit side by side on a frictionless floor: a 4.0 kg block on the left touching a 2.0 kg block on the right. You push the 4.0 kg block to the right with 24 N. Find the acceleration and the force each block exerts on the other.

    Show the solution
    1. Step 1: System = both blocks. The contact forces between them are internal and cancel. a=244.0+2.0=4.0a = \frac{24}{4.0 + 2.0} = 4.0 m/s² to the right.
    2. Step 2: System = the 2.0 kg block alone. The only horizontal force on it is the push from the 4.0 kg block: F=(2.0)(4.0)=8.0F = (2.0)(4.0) = 8.0 N to the right.
    3. Step 3: By the third law, the 2.0 kg block pushes back on the 4.0 kg block with 8.0 N to the left.
    4. Step 4: Check with the 4.0 kg block alone: net force = 24 − 8.0 = 16 N, and (4.0)(4.0) = 16 N. ✓

    Answer: a = 4.0 m/s²; the blocks push on each other with 8.0 N (rightward on the 2.0 kg block, leftward on the 4.0 kg block).

  2. Example 2

    Is it a pair? (classic trap)

    A 0.50 kg book sits at rest on a table. A student says, "The book's weight and the normal force are a Newton's third law pair, because they're equal and opposite." Explain what's wrong, and name the actual partner of each force.

    Show the solution
    1. Step 1: Third-law partners act on different objects. Both the weight and the normal force act on the book, so they can't be a pair.
    2. Step 2: They are different types of force: one gravitational, one a contact force from the table.
    3. Step 3: The weight's partner is the book pulling Earth upward with 4.9 N. The normal force's partner is the book pushing down on the table with 4.9 N.
    4. Step 4: The weight and the normal force are equal here only because the book is in equilibrium (Newton's first and second laws).

    Answer: They act on the same object, so they aren't a pair. Partners: the book's pull on Earth, and the book's push on the table.

Common mistakes

  • Thinking third-law pair forces cancel. They act on different objects, so they never appear on the same FBD.
  • Calling weight and normal force a pair. They act on the same object and are different types.
  • Believing the bigger or faster object exerts the bigger force in a collision. The forces are equal; the accelerations differ.
  • Using different tensions on the two sides of an ideal pulley. With an ideal string and pulley, the tension is the same throughout.

On the exam

  • Expect conceptual questions that compare the forces two objects exert on each other, such as a collision between unequal masses. The answer is always "equal"; the reasoning should name Newton's third law.
  • In multi-object problems, state which forces are internal to your chosen system. That shows the reader why they dropped out.

Connected topics

Videos

Check yourself

4 questions on 2.3 Newton’s Third Law. Pick an answer to see if you got it, and why.

Question 1 of 4Calculator allowed

A heavy truck collides head-on with a small car. How does the force the truck exerts on the car compare with the force the car exerts on the truck during the collision?

Question 2 of 4Calculator allowed

A horse pulls a cart forward and the cart speeds up. The cart pulls back on the horse with a force equal in size to the horse's pull. Why does the cart speed up anyway?

Question 3 of 4Calculator allowed

Earth exerts a 600 N gravitational force on a student standing on the ground. What is the third-law partner of this force?

Question 4 of 4Calculator allowed

An astronaut floats inside a spacecraft drifting through deep space, far from anything else. She pushes hard on the front wall. Which statement about the center of mass of the astronaut-plus-spacecraft system is true?

0 of 4 answered