Laws of Motion

• First: no net force ⟹ no change in velocity (rest or uniform straight-line motion).
• Second: $\mathbf{p}=m\mathbf{v}$, and

• Vector equation, applies per-component; local and instantaneous (no memory of past motion); for a system, $\mathbf{F}$ is the total external force only.
• Third: $\mathbf{F}_{AB} = -\mathbf{F}_{BA}$, simultaneous, on two different bodies — never a pair on the same body.

A large force acting for a short time — bat on ball, foot on ground — is just an ordinary force that happens to be brief:

Third law + second law, for two bodies A and B interacting over the same contact time:

Isolated system, no external force ⟹ total momentum constant — internal forces only shuffle it between particles. (Bullet + gun: zero total momentum before and after.)

• Two forces: $\mathbf{F}_1 = -\mathbf{F}_2$.
• Three or more concurrent forces:

Geometrically: forces drawn head-to-tail close into a polygon. In components, 3 independent scalar conditions, all must hold at once.

1. No sliding tendency: static friction $f_s = 0$.
2. Applied force below the limit: $f_s$ self-adjusts to exactly cancel it (body stays at rest).
3. At the ceiling: $f_s \le \mu_s N$, about to slip.
4. Sliding: friction drops to kinetic, $f_k = \mu_k N$, with $\mu_k$ reliably less than $\mu_s$.

Both coefficients depend only on the two surfaces in contact, never on contact area.

Only gravity and inter-molecular electrical forces are truly fundamental here. Normal reaction, tension, buoyancy, air resistance, friction, and the spring force $F=-kx$ are all electrical in origin, just expressed macroscopically.

Centripetal force is not a new force — just the name for whichever real force supplies the inward pull (tension, gravity, or friction):

Independent of the car’s mass.

At $v_0$, zero friction is needed — least tyre wear. Below it friction acts up the slope, above it (up to $v_{max}$) down the slope.