Newton's First and Second Laws

Newton's First Law (Law of Inertia)

Statement: An object at rest stays at rest, and an object in motion stays in motion at constant velocity, unless acted upon by a net external force.

$\text{If } \sum \vec{F} = 0, \text{ then } \vec{v} = \text{constant}$

Key Concepts

Inertia is the tendency of an object to resist changes in its motion.

• More massive objects have more inertia
• Inertia is measured by mass (kg)

Equilibrium occurs when the net force is zero.

• Static equilibrium: object at rest ($v = 0$)
• Dynamic equilibrium: object moving at constant velocity ($v = \text{constant}$, $a = 0$)

Common Misconceptions

❌ Wrong: "A force is needed to keep an object moving" ✓ Correct: A force is only needed to change motion (accelerate)

❌ Wrong: "If velocity is zero, force must be zero" ✓ Correct: Multiple forces can balance to give zero net force

Newton's Second Law

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

$\sum \vec{F} = m\vec{a}$

Or in component form: $\sum F_x = ma_x$ $\sum F_y = ma_y$

Understanding the Equation

Net force ($\sum \vec{F}$): Vector sum of all forces

• Measured in Newtons (N)
• $1 \text{ N} = 1 \text{ kg·m/s}^2$

Mass ($m$): Measure of inertia

• Measured in kilograms (kg)
• Scalar quantity (no direction)
• Not the same as weight!

Acceleration ($\vec{a}$): Rate of change of velocity

• Measured in m/s²
• Vector quantity (same direction as net force)
• If $\sum \vec{F} = 0$, then $a = 0$

Key Relationships

1. Direct relationship with force: If you double the force, you double the acceleration (same mass) $a \propto F \text{ (when } m \text{ is constant)}$

2. Inverse relationship with mass: If you double the mass, you halve the acceleration (same force) $a \propto \frac{1}{m} \text{ (when } F \text{ is constant)}$

3. Direction: Acceleration is always in the same direction as the net force

Mass vs. Weight

Mass

• Mass is the amount of matter in an object
• Scalar quantity
• Measured in kg
• Same everywhere in the universe
• Measure of inertia

Weight

• Weight is the gravitational force on an object
• Vector quantity (points toward Earth's center)
• Measured in Newtons (N)
• Depends on location (gravity varies)
• Formula: $W = mg$

On Earth's surface: $W = mg \quad \text{where } g = 9.8 \text{ m/s}^2$

On the Moon: $g_{moon} \approx 1.6$ m/s² (about 1/6 of Earth)

• Same mass, but weight is 1/6 as much!

Free Body Diagrams (FBDs)

A free body diagram shows all forces acting on a single object.

Steps to Draw an FBD

1. Isolate the object (draw a dot or box)
2. Draw all forces as arrows starting from the object
   • Length represents magnitude
   • Direction shows force direction
3. Label each force (e.g., $F_N$, $F_g$, $F_T$)
4. Do NOT include:
   • Motion (velocity, acceleration)
   • Forces the object exerts on other things

Common Forces

• Weight ($\vec{F}_g$ or $\vec{W}$): Always points downward, magnitude $mg$
• Normal force ($\vec{F}_N$ or $\vec{N}$): Perpendicular to surface, pushes away from surface
• Tension ($\vec{F}_T$ or $\vec{T}$): Along rope/string, pulls toward rope
• Friction ($\vec{f}$): Parallel to surface, opposes motion/attempted motion
• Applied force ($\vec{F}_{app}$): Push or pull from external agent

Problem-Solving Strategy

1. Draw a free body diagram
2. Choose coordinate system (align with motion when possible)
3. Write $\sum F_x = ma_x$ and $\sum F_y = ma_y$
4. List all forces in each direction (use + and - signs)
5. Solve for unknowns
6. Check units and reasonableness

Newton's Laws Summary

| Law | Statement | Equation | |-----|-----------|----------| | First | Object maintains velocity unless net force acts | $\sum \vec{F} = 0 \Rightarrow \vec{v} = \text{const}$ | | Second | Net force causes acceleration | $\sum \vec{F} = m\vec{a}$ | | Third | Forces come in equal/opposite pairs | $\vec{F}_{AB} = -\vec{F}_{BA}$ |

Special Cases

Zero Acceleration

If $a = 0$, then $\sum F = 0$ (equilibrium)

• Object at rest OR moving at constant velocity
• All forces balance

Constant Acceleration

If $a = \text{constant}$, then $\sum F = \text{constant}$

• Can use kinematic equations
• Common in free fall, inclined planes

Variable Acceleration

If $a$ changes, then $\sum F$ changes

• More complex analysis required
• Need calculus or numerical methods