Newton’s Laws of Motion & Friction
Competitive‑physics (IIT‑JEE) Study Notes
Goal – Build a rigorous, exam‑oriented understanding of Newton’s laws together with the mechanics of friction.
Structure –
- Quick recap of the three laws
- Types of friction & their governing equations
- Detailed derivations (incl. limiting cases)
- Important formulae & “tricky” points
- A hard practice question + hint
1. Quick Recap of Newton’s Laws
| Law | Mathematical Statement | Key Insight |
|---|---|---|
| 1st | ( \sum \mathbf{F} = \mathbf{0} ) ⇔ ( \mathbf{a}=0 ) | A body remains at rest or moves with constant velocity unless acted on by a net external force. |
| 2nd | ( \sum \mathbf{F} = m\mathbf{a} ) | The net external force equals mass times acceleration. The vector form is essential for exam problems. |
| 3rd | ( \mathbf{F}{AB} = -\mathbf{F}{BA} ) | Action–reaction pairs are equal in magnitude, opposite in direction. |
Tip – Always draw the free‑body diagram (FBD) first. Label all forces, including the normal, weight, friction, tension, applied forces, etc. Then apply the 2nd law component‑wise.
2. Types of Friction & Governing Equations
| Category | Formula | Conditions | Notes |
|---|---|---|---|
| Static Friction | ( f_s \le \mu_s N ) | No relative motion at the interface. | ( \mu_s ) is the coefficient of static friction. |
| Kinetic (Sliding) Friction | ( f_k = \mu_k N ) | Relative motion present. | ( \mu_k < \mu_s ) generally. |
| Rolling Friction | ( f_{\text{roll}} = \mu_{\text{roll}} N ) | Rolling without slipping. | ( \mu_{\text{roll}} ) is typically (\ll \mu_k). |
| Viscous Friction | ( f_v = b v ) | Fluid or gas drag; linear with velocity (low (Re)). | At high (Re) it becomes (f \propto v^2). |
| Coulomb Friction | Same as static/kinetic above. | Commonly used to denote dry friction. |
Important – The static friction adjusts up to its maximum value; it is not a constant. In equilibrium problems, treat it as an unknown (f_s) bounded by (\mu_s N).
3. Detailed Derivations
3.1 Static vs Kinetic Friction – Limiting Process
Consider a block on a horizontal surface with applied horizontal force (F).
- Static regime: (f_s = F) (as long as (F \le \mu_s N)).
- At threshold: (F = \mu_s N).
- Beyond threshold: Block begins to move, kinetic friction takes over: (f_k = \mu_k N).
Derivation of (f_s^{\max} = \mu_s N)
- Assume two blocks in contact (block A sliding on block B).
- The maximum static friction equals the maximum static friction attainable at the interface: (f_{s,AB}^{\max} = \mu_s N_{AB}).
- Transfer the force through the interface: (f_{s,AB}^{\max}) is what the upper block can exert on the lower before slipping.
- Hence (f_s^{\max}) is the limiting case of static friction.
Tricky Point – In problems, you cannot simply set (f_s = \mu_s N) unless you are explicitly asked for the maximum static friction. Usually, you treat (f_s) as an unknown and use the inequality.
3.2 Friction on an Inclined Plane
Let a block of mass (m) be on a plane inclined at angle (\theta) to the horizontal.
- Weight components: (mg \sin\theta) (down the plane), (mg \cos\theta) (normal).
- Normal force: (N = mg \cos\theta).
- Static friction: (f_s \le \mu_s N = \mu_s mg \cos\theta).
Condition for impending motion up the plane
( mg \sin\theta - f