Law in Words

Newton’s Second Law

The rate of change of momentum of a body is directly proportional to the external force applied and takes the direction of that force.

Variables: Force (F), momentum \(p = m v\), and time interval \(\Delta t\). Why does “rate of change” point to acceleration rather than speed?

Equation Reveal

\[F = m\,a\]

Variable Definitions

F Force, \(N\)
m Mass, \(kg\)
a Acceleration, \(m/s^{2}\)

Applications

Label & Learn

Drag each unit label onto F, m, and a.

Unit Check

Confirm \(N = kg·m/s^{2}\) so both sides match.

Quick Example

If \(m = 2\,kg\) and \(a = 3\,m/s^{2}\), then \(F = 6\,N\).

Source: CBSE Grade 11 Textbook

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Force vs Acceleration

Force vs acceleration graph

Force (N) vs Acceleration (m/s²)

Reading the F–a Graph

Linear relationship: a straight line through the origin shows force is directly proportional to acceleration for a fixed mass.

Graph interpretation: the slope \( \frac{F}{a} \) equals the mass. Steeper slope ⇒ heavier body.

Key Points:

  • Slope gives mass in kilograms.
  • Doubling force doubles acceleration when the mass is constant.
  • Quiz: Read the slope—mass here is 2 kg.
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From p to F = m a

1
\[\vec{p}=m\vec{v}\]

Linear momentum equals mass times velocity.

2
\[\vec{F}=\frac{d\vec{p}}{dt}\]

Force is the time rate of change of momentum.

3
\[\frac{d\vec{p}}{dt}=m\frac{d\vec{v}}{dt}\]

Assuming \(m\) is constant, only velocity varies with time.

4
\[\vec{F}=m\vec{a}\]

Since \( \frac{d\vec{v}}{dt}=\vec{a} \), we reach the familiar form.

Key Insight:

Constant mass lets the momentum derivative collapse to acceleration, revealing \( \vec{F}=m\vec{a} \).

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Multiple Choice Question

Question

A 3 kg drone experiences a net forward force of 12 N. What is its acceleration?

1
2 m/s²
2
3 m/s²
3
4 m/s²
4
6 m/s²

Hint:

Apply \(a = \frac{F}{m}\).

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Key Takeaways

Proportionality

Force increases directly with acceleration: \(F \propto a\).

Vector Direction

\( \vec F \) and \( \vec a \) point the same way—both are vectors.

Mass from Slope

Slope of an \(F\)–\(a\) graph equals the object’s mass.

Momentum Link

Same law: \(F = \\frac{dp}{dt}\)—force is rate of momentum change.