Forces and Newton's Laws
Forces and Newton's Laws
Newton's laws of motion form the foundation of classical mechanics. At A-Level, you must be able to model physical situations, resolve forces, and apply F = ma to find accelerations, tensions, and reactions.
Newton's Three Laws
First Law (Inertia): A body remains at rest or continues to move at constant velocity unless acted upon by a resultant (net) force.
Second Law: The resultant force on a body equals its mass times its acceleration:
F = ma (force in newtons, mass in kg, acceleration in m/s²)
This is a vector equation — it applies independently in each direction.
Third Law: When two bodies interact, the forces they exert on each other are equal in magnitude and opposite in direction (action-reaction pairs). These forces act on different bodies.
Types of Force
| Force | Symbol | Description |
|---|---|---|
| Weight | W = mg | Gravitational force, always acts downward |
| Normal reaction | R or N | Contact force perpendicular to surface |
| Tension | T | Force in a taut string or rope |
| Friction | F | Opposes relative motion along the surface |
| Thrust/push | P | Applied force |
| Air resistance/drag | D | Opposes motion through air |
Resolving Forces
When forces act at angles, resolve into perpendicular components (usually horizontal and vertical):
- Horizontal component = Fcosθ (where θ is measured from the horizontal)
- Vertical component = Fsinθ
Then apply F = ma separately in each direction.
Worked Example: A 5 kg block is pulled along a smooth horizontal surface by a force of 30 N at 40° above the horizontal. Find the acceleration and the normal reaction.
Horizontally (→ positive): 30cos40° = 5a
a = 30cos40°/5 = 22.98/5 = 4.60 m/s²
Vertically (↑ positive): R + 30sin40° - 5g = 0
R = 5(9.8) - 30sin40° = 49 - 19.28 = 29.7 N
Connected Particles
When particles are connected by a string (possibly over a pulley), they have the same magnitude of acceleration (assuming inextensible string). Apply F = ma to each particle separately, then solve simultaneously.
Worked Example: Two particles of mass 3 kg and 5 kg are connected by a light inextensible string over a smooth pulley (Atwood's machine). Find the acceleration and the tension.
For the 5 kg mass (taking downward as positive):
5g - T = 5a ... (i)
For the 3 kg mass (taking upward as positive, since it accelerates upward):
T - 3g = 3a ... (ii)
Adding (i) and (ii): 5g - 3g = 8a
2g = 8a, so a = g/4 = 2.45 m/s²
From (ii): T = 3(2.45) + 3(9.8) = 7.35 + 29.4 = 36.75 N
Inclined Planes
On a plane inclined at angle θ to the horizontal, resolve forces parallel and perpendicular to the slope:
- Component of weight down the slope = mgsinθ
- Component of weight into the slope = mgcosθ
- Normal reaction R = mgcosθ (if no other forces act perpendicular to the slope)
Worked Example: A 4 kg block slides down a smooth slope inclined at 30°. Find the acceleration.
Parallel to slope (down positive): mgsin30° = ma
a = gsin30° = 9.8 × 0.5 = 4.9 m/s²
Particles on Slopes Connected by Strings
Worked Example: A 2 kg block on a smooth slope at 30° is connected by a string over a pulley at the top of the slope to a 3 kg block hanging vertically. Find the acceleration.
For the 3 kg block (downward positive): 3g - T = 3a ... (i)
For the 2 kg block (up the slope positive): T - 2gsin30° = 2a ... (ii)
Adding: 3g - 2gsin30° = 5a
3(9.8) - 2(9.8)(0.5) = 5a
29.4 - 9.8 = 5a
a = 19.6/5 = 3.92 m/s²
Equilibrium
A body is in equilibrium if the resultant force is zero in every direction (acceleration = 0). For a body in equilibrium:
- Sum of forces horizontally = 0
- Sum of forces vertically = 0
Modelling Assumptions
Questions often use phrases that indicate simplifications:
| Phrase | Meaning |
|---|---|
| "Light string" | String has zero mass (tension is the same throughout) |
| "Inextensible string" | String does not stretch (particles have same acceleration) |
| "Smooth surface" | No friction |
| "Particle" | No size, no rotation — all forces act at one point |
| "Smooth pulley" | No friction at the pulley (tension same on both sides) |
Exam Tips
- Always draw a force diagram for each body. Label every force clearly.
- Apply F = ma to each body separately — never mix forces on different bodies in one equation.
- On inclined planes, resolve parallel and perpendicular to the slope, not horizontally and vertically.
- In connected particle problems, define a consistent positive direction for all bodies (e.g., the direction of acceleration).
- Newton's Third Law pairs act on different bodies. If you include the weight of a book on a table (Earth pulls book down), the reaction pair is the book pulling the Earth up — not the normal reaction from the table.