Comprehensive theory, key formulas, diagrams, and memory aids for Dynamics.
Newton's First Law states that an object will remain at rest or continue to move in a straight line at constant velocity unless acted upon by a resultant external force. This property of matter — its resistance to changes in its state of motion — is called inertia. The greater an object's mass, the greater its inertia.
Practical consequences: A passenger lurches forward when a bus brakes because their body tends to continue at the previous velocity. A tablecloth can be pulled from under dishes because the dishes' inertia keeps them stationary during a rapid pull.
The resultant force acting on an object equals the rate of change of its momentum:
$$F = \frac{\Delta p}{\Delta t} = \frac{\Delta(mv)}{\Delta t}$$
For constant mass, this simplifies to the familiar $F = ma$, where: - $F$ = resultant force (N) - $m$ = mass (kg) - $a$ = acceleration (m s⁻²)
The direction of acceleration is always the same as the direction of the resultant force. Note that force and acceleration are vectors — when multiple forces act, you must use vector addition to find the resultant before applying $F = ma$.
Free body diagrams are essential tools that show all forces acting on an object as labelled arrows. Each force arrow should start from the centre of mass, point in the direction of the force, and have a length proportional to its magnitude.
Newton's Third Law: For every action force, there is an equal and opposite reaction force, acting on the other body. The action-reaction pair: - Are equal in magnitude - Act in opposite directions - Are of the same type (e.g., both gravitational, or both contact forces) - Act on different objects
Example: When a rocket expels hot gases backward (action), the gases push the rocket forward (reaction). When you stand on the Earth, you push down on the Earth (action) and the Earth pushes up on you via the normal contact force (reaction).
Common misconception: Newton's Third Law pairs cannot cancel each other because they act on different objects. Forces cancel only when they act on the same object.
The linear momentum of an object is: $$\vec{p} = m\vec{v}$$
Momentum is a vector (same direction as velocity) measured in kg m s⁻¹.
The principle of conservation of momentum states that the total linear momentum of a closed system (no external forces) remains constant. This applies to all collisions and explosions.
$$m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$$
| Type | Kinetic Energy | Momentum |
|---|---|---|
| Elastic | Conserved | Conserved |
| Inelastic | Lost (as heat, sound) | Conserved |
| Perfectly inelastic | Maximum loss | Conserved (objects stick together) |
| Explosion | Increases (from stored energy) | Conserved (total = 0 if initially at rest) |
In an explosion from rest, since total initial momentum is zero, the two objects must move apart with equal and opposite momenta: $m_1 v_1 = -m_2 v_2$.
Impulse is the change in momentum produced by a force acting over a time interval: $$J = F\Delta t = \Delta p = mv - mu$$
Impulse has units of N s (equivalent to kg m s⁻¹). The impulse-momentum theorem is a direct consequence of Newton's Second Law.
On a force–time graph, the area under the graph equals the impulse. This is particularly useful when the force is not constant (e.g., during a collision).
The concept of impulse explains why safety features like airbags and crumple zones reduce injury: they increase the time of collision ($\Delta t$), which reduces the average force ($F$) for the same change in momentum.
graph LR
A[Force applied] --> B[Acts over time Δt]
B --> C[Impulse = FΔt]
C --> D[Change in momentum Δp = mv - mu]
D --> E[Object changes velocity]
On the Moon ($g \approx 1.6$ m s⁻²), an object's weight is 1/6 of its Earth weight, but its mass is unchanged.
Friction is a contact force opposing relative motion (or tendency of motion) between surfaces. It arises from electromagnetic interactions between surface atoms.
An object is in mechanical equilibrium when the resultant force is zero (and resultant torque is zero). A static object in equilibrium satisfies: $$\sum F_x = 0, \quad \sum F_y = 0$$
For three forces in equilibrium, they can be represented as the three sides of a closed triangle (Lami's theorem). For multiple forces, resolve all forces into horizontal and vertical components and equate sums to zero.