Newton’s Laws of Motion

Newton’s three laws of motion, formulated by Sir Isaac Newton in his seminal work *Philosophiæ Naturalis Principia Mathematica* (1687), form the foundation of classical mechanics. These laws describe how forces govern the motion of objects, from everyday scenarios to celestial dynamics. Integrating concepts like inertia, force, acceleration, momentum, work, and energy, this MathMultiverse guide provides detailed explanations, mathematical formulations, examples, and applications, enhanced with interactive visualizations.

First Law: Inertia

An object at rest remains at rest, and an object in motion continues in a straight line at constant velocity unless acted upon by a net external force. This property, known as inertia, depends on mass—the greater the mass, the more inertia.

Key Concept: Inertia is quantified by mass \( m \) (in kg), with no specific equation for the law itself.

Second Law: F = ma

The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass:

\[ F = m \cdot a \]

Where:

  • \( F \): Net force (in Newtons, N)
  • \( m \): Mass (in kg)
  • \( a \): Acceleration (in m/s²)

Related Formulas:

  • Weight: \( W = m \cdot g \), where \( g \approx 9.8 \, \text{m/s}^2 \)
  • Momentum: \( p = m \cdot v \), where \( v \) is velocity (in m/s)

Third Law: Action-Reaction

For every action, there is an equal and opposite reaction:

\[ F_{\text{action}} = -F_{\text{reaction}} \]

Forces act in pairs with equal magnitude and opposite direction.

Examples

Let’s apply Newton’s Laws with detailed calculations.

Stationary Object (First Law)

A 5 kg box on a frictionless table remains at rest until a net force is applied, demonstrating inertia.

Moving Object (First Law)

A hockey puck sliding at 2 m/s on ice continues until friction intervenes.

Force and Acceleration (Second Law)

A 2 kg box is pushed with 10 N:

\[ F = m \cdot a \] \[ 10 = 2 \cdot a \] \[ a = \frac{10}{2} = 5 \, \text{m/s}^2 \]

Weight Calculation (Second Law)

Weight of a 3 kg object (\( g = 9.8 \, \text{m/s}^2 \)):

\[ W = m \cdot g \] \[ W = 3 \cdot 9.8 = 29.4 \, \text{N} \]

Momentum (Second Law)

Momentum of a 5 kg ball at 4 m/s:

\[ p = m \cdot v \] \[ p = 5 \cdot 4 = 20 \, \text{kg·m/s} \]

Jumping (Third Law)

Jumping exerts a downward force, with the ground pushing upward equally.

Rocket Propulsion (Third Law)

A rocket expels gas downward, receiving an equal upward force.

Force vs. Acceleration

Linear relationship for a 2 kg mass: \( F = 2 \cdot a \).

Applications

Newton’s Laws underpin numerous physical phenomena.

Work

Work done by a force over a displacement:

\[ W = F \cdot d \cdot \cos\theta \]

Car Acceleration

A 1000 kg car accelerates at 2 m/s², with force:

\[ F = m \cdot a \] \[ F = 1000 \cdot 2 = 2000 \, \text{N} \]

Work over 10 m (\( \theta = 0^\circ \)):

\[ W = F \cdot d \cdot \cos 0^\circ \] \[ W = 2000 \cdot 10 = 20000 \, \text{J} \]

Kinetic Energy

Kinetic energy of a 2 kg ball at 3 m/s:

\[ KE = \frac{1}{2} m v^2 \] \[ KE = \frac{1}{2} \cdot 2 \cdot 3^2 = 9 \, \text{J} \]

Potential Energy

Potential energy of a 5 kg object at 10 m (\( g = 9.8 \, \text{m/s}^2 \)):

\[ PE = m \cdot g \cdot h \] \[ PE = 5 \cdot 9.8 \cdot 10 = 490 \, \text{J} \]

Elevator Motion

Tension in an 800 kg elevator cable accelerating upward at 1.5 m/s²:

\[ W = m \cdot g = 800 \cdot 9.8 = 7840 \, \text{N} \] \[ F_{\text{net}} = m \cdot a = 800 \cdot 1.5 = 1200 \, \text{N} \] \[ T - W = F_{\text{net}} \] \[ T = 7840 + 1200 = 9040 \, \text{N} \]

Friction and Work

Work on a 10 kg box pushed 5 m with 20 N against 5 N friction:

\[ W_{\text{net}} = F \cdot d \cdot \cos 0^\circ - F_{\text{friction}} \cdot d \] \[ W_{\text{net}} = 20 \cdot 5 - 5 \cdot 5 = 100 - 25 = 75 \, \text{J} \]

Rocket Launch

Thrust for a 2000 kg rocket accelerating at 3 m/s²:

\[ W = m \cdot g = 2000 \cdot 9.8 = 19600 \, \text{N} \] \[ F_{\text{net}} = m \cdot a = 2000 \cdot 3 = 6000 \, \text{N} \] \[ T = W + F_{\text{net}} = 19600 + 6000 = 25600 \, \text{N} \]

Kinetic Energy vs. Velocity

Kinetic energy for a 2 kg mass: \( KE = \frac{1}{2} \cdot 2 \cdot v^2 \).