Unit 2 · Topic 03

Newton's second law

The second law relates the sum of external forces to a system's acceleration. The relation is vectorial and requires an explicit boundary and axes.

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Concept 01

Net force and acceleration

Essential The minimum you should retain

For constant mass, ΣF→ext=ma→ \sum\vec F_{ext}=m\vec a . Acceleration points along net force, not necessarily along velocity.

UnderstandInterpret and connect

The equation uses the resultant of all external forces. An individual force equals ma only when it is the sole external contribution.

DeepenFormulation and conditions

Because m is a positive scalar, every component of a is proportional to the corresponding component of ΣF_ext.

ExploreConnections for further study

A net force perpendicular to velocity mainly changes its direction; a parallel component changes speed.

Mathematical relation

Newton's second law for constant mass

ΣF→ext=ma→ \sum\vec F_{ext}=m\vec a
Represents

The relation between external net force and acceleration of the same system.

Physical interpretation

Acceleration is parallel to net force and has magnitude F_net/m.

DeepenVariables, conditions, and checks

Variables

m
inertial mass of the system; usual unit: kg
a⃗
acceleration of the system; usual unit: ms² m/s^2

Conditions of application

  • Mass is constant.
  • Forces and acceleration are described in an inertial frame.

Dimensional check

N = kg· ms² m/s^2 .

Errors it helps prevent

  • Equating one individual force with ma.
  • Assuming acceleration and velocity must be parallel.
From a particle, velocity points right while net force and acceleration point upward.vΣF_extasystem

Net force and acceleration point north. Instantaneous velocity may point east: force changes the velocity vector.

Concept 02

Component equations

Essential The minimum you should retain

One vector equation can be written as ΣFx=max \sum F_x=ma_x , ΣFy=may \sum F_y=ma_y , and ΣFz=maz \sum F_z=ma_z .

UnderstandInterpret and connect

Signs come from the chosen axes. Choose axes that make the geometry and unknowns clear.

DeepenFormulation and conditions

Components are projections of one vector equation; they are not new forces. Solving them separately preserves the full physical relation.

ExploreConnections for further study

A different basis changes components but not the physical net-force or acceleration vectors.

Mathematical relation

Newton's second law in Cartesian components

ΣFx=max,ΣFy=may,ΣFz=maz \sum F_x=ma_x,\quad\sum F_y=ma_y,\quad\sum F_z=ma_z
Represents

Three scalar projections of one vector equation.

Physical interpretation

Each axis can be solved separately without creating extra forces.

DeepenVariables, conditions, and checks

Variables

ΣFᵢ
component i of net force; usual unit: N
aᵢ
component i of acceleration; usual unit: ms² m/s^2
m
inertial mass; usual unit: kg

Conditions of application

  • Components use the same basis and sign convention.

Dimensional check

Every component is expressed in N.

Errors it helps prevent

  • Treating components as new forces.

Concept 03

How force, mass, and acceleration change

Essential The minimum you should retain

At fixed mass, doubling net force doubles acceleration. At fixed net force, doubling mass halves acceleration.

UnderstandInterpret and connect

The graph of a against F_net is a straight line through the origin with slope 1/m. The graph of a against m is an inverse curve for fixed F_net.

DeepenFormulation and conditions

The proportionalities a ∝ F_net and a ∝ 1/m compare situations only when the other stated conditions remain fixed.

ExploreConnections for further study

Measuring several force-acceleration pairs makes it possible to estimate inertial mass from the slope.

a against F_net for m = 2 kg

Line of acceleration equal to net force divided by two kilograms.
  • a=F_net/2

The line passes through the origin with slope 1/m = 0.5 kg⁻¹: doubling F_net doubles a.

a against m for F_net = 12 N

Decreasing curve of acceleration equal to twelve newtons divided by mass.
  • a=12/m

The inverse curve shows that increasing mass reduces acceleration when net force remains fixed.

Concept 04

Scope of the form ΣF = ma

Essential The minimum you should retain

This course uses ΣF→ext=ma→ \sum\vec F_{ext}=m\vec a for constant-mass systems.

UnderstandInterpret and connect

Mass, net force, and acceleration must refer to the same system and the same modeled instant or interval.

DeepenFormulation and conditions

The more general form connects external net force with the rate of change of linear momentum: ΣF→ext=dp→dt \sum\vec F_{ext}=d\vec p/dt .

ExploreConnections for further study

For variable-mass systems, ma must not be applied automatically. Careful treatment of matter exchange belongs to a later unit.

Concept review

Common errors

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

Forcing acceleration to point along velocity.

Acceleration points with net force; compare it with v only to interpret how motion changes.

Applying ma to one force while ignoring the others.

Choose the system, add every external force, and apply the second law to the resultant.