Newton’s Laws (Vectors & Problem Solving)

1. Definitions (Must Know)

A. Mass and inertia

Mass is the property of a body that resists a change in motion. This resistance is called inertia. A larger resultant force is required to give a larger mass the same acceleration.

B. Newton’s first law

If the resultant external force on a body is zero, its velocity remains constant:

∑ vector Fₑₓₜ = 0 ⇒ vector a = 0 ⇒ vector v = constant

C. Newton’s second law

Newton’s 2nd law is written most generally as:

∑ vector Fₑₓₜ = (d vector p)/dt

where momentum is: vector p = m vector v

See: Momentum.

For a constant mass system (most A Level mechanics problems), this reduces to:

∑ vector F = m vector a

D. Newton’s third law (interaction pair)

If body A exerts a force on body B, then body B exerts an equal and opposite force on body A:

vector F_(A on B) = - vector F_(B on A)

2. Key Ideas (What Earns Marks)

• Always start with an FBD for the body you are analysing: Free Body Diagrams.
• Choose axes and a sign convention before writing equations (e.g. +x along the motion).
• Resolve forces only when needed: Vector Addition & Components.
• Apply Newton’s 2nd law component-by-component:
  • ∑ Fₓ = maₓ
  • ∑ F_y = ma_y
• Action–reaction pairs act on different bodies (never draw both forces on the same FBD).
• For a point in equilibrium under exactly three forces, a scaled head-to-tail vector triangle must close.

3. Detailed Explanations

A. The exam workflow (FBD → components → equations)

1. Choose the body (one object at a time).
2. Draw its FBD with all external forces.
3. Choose axes and write down your sign convention.
4. Resolve forces into components (if the forces are not already along your axes).
5. Write ∑ Fₓ = maₓ and ∑ F_y = ma_y.
6. Solve, then check direction/signs/units.

B. Component form (what you actually write)

If you choose +x to the right and +y upwards, typical forces look like this:

• weight: vector W = (0, -mg)
• normal reaction on a horizontal surface: vector N = (0, + N)
• friction (opposes motion): direction depends on motion/impending motion

C. Newton’s 2nd law: F = ma is a special case

Newton’s 2nd law is defined using momentum:

∑ vector Fₑₓₜ = (d vector p)/dt

If mass is constant, vector p = m vector v and:

(d vector p)/dt = (d(m vector v))/dt = m(d vector v)/dt = m vector a

So you may use:

∑ vector F = m vector a

D. Vector triangles in equilibrium

For exactly three coplanar forces on a point mass in equilibrium, draw the force vectors head-to-tail using a stated scale. The third arrow must finish at the starting point because their vector sum is zero. A force triangle represents force directions and magnitudes; it is not a sketch of the apparatus.

4. Common Mistakes

• Putting an action–reaction pair on the same FBD (they act on different bodies).
• Writing N = mg automatically (false on slopes, in lifts, or when there is vertical acceleration).
• Choosing axes, then forgetting to apply signs consistently (especially on slopes).
• Mixing “resultant force” in as if it is an extra force (it is the sum of forces).

5. Exam Tips

• Start with: “take +x along the motion” (or another clear choice).
• If a surface is smooth: set friction = 0 (but still include N).
• If speed is constant: vector a = 0 so ∑ vector F = 0 (but you still need the FBD).
• For connected bodies: write one ∑ F = ma equation per body, then use the string constraint (same acceleration if string is taut).

6. Worked Examples

7. Mind Stretchers