Key Points

Aristotle incorrectly believed an external force is needed to keep a body in motion. Galileo corrected this, establishing the law of inertia: an object resists changes to its state of rest or uniform motion, a property called inertia.

A body remains at rest or in uniform motion in a straight line unless acted upon by a net external force. If the net external force on a body is zero ($\sum \mathbf{F} = 0$), its acceleration is zero ($\mathbf{a} = 0$).

Momentum ($\mathbf{p}$) is the product of a body's mass ($m$) and velocity ($\mathbf{v}$). It is a vector quantity defined by the formula $\mathbf{p} = m\mathbf{v}$. Its SI unit is $\text{kg m/s}$.

The rate of change of momentum of a body is directly proportional to the applied net external force. This is expressed as $\mathbf{F} = \frac{d\mathbf{p}}{dt}$, which simplifies to $\mathbf{F} = m\mathbf{a}$ for a constant mass. The SI unit of force is the newton (N).

Impulse is the product of force and the time duration for which it acts, and it equals the change in momentum of the body. Impulse = $\mathbf{F} \Delta t = \Delta \mathbf{p}$. It is useful for large forces acting over a short time.

To every action, there is always an equal and opposite reaction. Forces always occur in pairs, acting on different bodies; $\mathbf{F}_{AB} = -\mathbf{F}_{BA}$. Action and reaction forces cannot cancel each other out.

The total momentum of an isolated system (a system with no net external force) remains constant. If two bodies collide, the total momentum before collision equals the total momentum after collision.

A particle is in equilibrium when the net external force acting on it is zero. For a particle under the action of multiple forces $\mathbf{F}_1, \mathbf{F}_2, \mathbf{F}_3, ...$, the condition is $\sum \mathbf{F} = \mathbf{F}_1 + \mathbf{F}_2 + \mathbf{F}_3 + ... = 0$.

Static friction ($f_s$) is a self-adjusting force that opposes the impending motion between surfaces in contact. Its value ranges from zero up to a maximum limit, given by $(f_s)_{\text{max}} = \mu_s N$, where $\mu_s$ is the coefficient of static friction and N is the normal reaction.

Kinetic friction ($f_k$) opposes the relative motion between surfaces in contact and has a constant value. It is given by the formula $f_k = \mu_k N$, where $\mu_k$ is the coefficient of kinetic friction. Generally, $\mu_k < \mu_s$.

Centripetal force is the net force required to keep a body of mass $m$ moving in a circle of radius $R$ with speed $v$. It is always directed towards the center of the circle and its magnitude is $F_c = \frac{mv^2}{R}$.

For a car on a level road, the centripetal force is provided by static friction. The maximum safe speed to avoid skidding is $v_{\text{max}} = \sqrt{\mu_s R g}$.

Banking a road at an angle $\theta$ helps provide the necessary centripetal force. The optimum speed to avoid wear and tear on tires (no friction needed) is $v_o = \sqrt{R g \tan\theta}$.

A free-body diagram is essential for solving mechanics problems. It is a diagram showing a chosen body isolated from its surroundings, with all the external forces acting on it represented by vectors.