Unit 3 · Topic 02

Particle dynamics

In dynamics the net force is no longer zero: its direction and magnitude determine acceleration. The main difficulty is often modelling interactions and constraints correctly before solving the equations.

Unit 3DynamicsOpen navigation

Concept 01

From the FBD to the equation

Essential The minimum you should retain

The basic procedure is: choose the system, draw the FBD, select axes, sum force components, and apply ΣF→ext=ma→ \sum\vec F_{ext}=m\vec a .

UnderstandInterpret and connect

Acceleration must refer to the same system as the forces. If the system is a box, its equation cannot contain a force that acts on the rope or another block.

DeepenFormulation and conditions

For constant mass, each component satisfies ΣF_i = ma_i. One component may be in equilibrium while another accelerates; for example a_y = 0 while a_x ≠ 0.

ExploreConnections for further study

Separating modelling from algebra makes the same structure reusable across many problems. Geometry changes components, not the underlying dynamical balance.

Concept 02

Multi-body systems

Essential The minimum you should retain

When several bodies interact, analyse them separately or choose them as a single system. The best choice depends on the unknown you need.

UnderstandInterpret and connect

Separate-body analysis exposes internal forces such as tension or contact. Grouping the bodies can remove those internal forces from the external balance, making the common acceleration easier to find.

DeepenFormulation and conditions

If masses m1 and m2 move together under a horizontal force F and other horizontal external forces are negligible, the combined system satisfies F = (m1+m2)a. A second equation for one block can then determine the internal contact force.

ExploreConnections for further study

A larger system simplifies some unknowns while hiding others. This freedom is a modelling tool: there is not one universal FBD for a scene, but one correct FBD for each declared system.

Worked example

Two blocks accelerating together

Two 2 kg and 3 kg blocks touch on a frictionless horizontal surface. A 20 N force pushes the first block toward the second.

Given
  • m1 = 2 kg
  • m2 = 3 kg
  • F = 20 N
Target

Find the common acceleration and the contact force on m2.

  1. Combined system

    For both blocks, a = 20/(2+3) = 4 m/s².

  2. Isolate m2

    The only horizontal force on m2 is contact C.

  3. Second law

    C = m2a = 3(4) = 12 N.

  4. Interaction pair

    The third-law force on m1 has magnitude 12 N in the opposite direction.

Conclusion

The common acceleration is 4 m/s² and the contact force on m2 is 12 N.

Two touching blocks lie inside a combined boundary; another boundary isolates the second block while an external force pushes the first.system 1+2system 2m₁m₂external Fcontact on 2

The small boundary isolates block 2 and shows contact. The large boundary includes both blocks, so contact is internal and leaves the external balance.

Concept 03

Motion constraints

Essential The minimum you should retain

Connected bodies cannot always accelerate independently. An inextensible rope or contact surface may impose a relation between their motions.

UnderstandInterpret and connect

For a simple ideal rope over a fixed pulley, motion of one end must be matched by the other end so the total rope length remains constant.

DeepenFormulation and conditions

Geometric constraints provide extra kinematic equations. For one inextensible rope over an ideal fixed pulley, the acceleration magnitudes of the ends are equal, although signs depend on the chosen axes.

ExploreConnections for further study

Movable-pulley systems can produce relations such as 2a_A + a_B = 0. Such relations should be derived from the total rope length rather than memorized.

Concept 04

Physical checks

Essential The minimum you should retain

After solving, ask whether the acceleration direction and force magnitudes make physical sense.

UnderstandInterpret and connect

An algebraic result may be mathematically valid but incompatible with the assumed regime. A negative normal force may indicate loss of contact; a required static friction larger than μ_sN means the assumed rest state cannot persist.

DeepenFormulation and conditions

Limiting cases help validate expressions. If an applied force tends to zero, the solution should approach the corresponding unforced behaviour. If mass becomes very large under the same net force, acceleration should decrease.

ExploreConnections for further study

These checks are part of scientific modelling. Equations do more than produce numbers: they also reveal when an assumption becomes inconsistent.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Mixing forces and accelerations from different bodies in one equation.

Each Newton equation must refer to one declared system.

Guessing a pulley acceleration relation instead of deriving it.

Write the constant rope length and derive the constraint.