Newton’s Laws of Motion

Newton's First Law functions as both:

• A physical law describing natural behavior
• A definition of inertial reference frames

Key insight: The law states that unaccelerated motion occurs if and only if no net force acts, but this only holds true in certain reference frames (inertial frames).

Since we define inertial frames as "those where Newton's First Law holds true," the law becomes circular when universalized across all reference frames.

In practice: We identify an approximately inertial frame (like Earth) and verify the law works well enough for most applications.

The person would observe:

• Zero acceleration of the lamp relative to the cabin
• The lamp appears stationary in the cabin frame

This demonstrates that Newton's First Law gives contradictory results when applied from different frames:

• From cabin frame: Zero acceleration implies net force is zero (W = T)
• From ground frame: Downward acceleration of 9.8 m/s² implies net force is not zero (W ≠ T)

This contradiction proves that at least one frame must be non-inertial. In this case, the freely falling elevator is non-inertial because it's accelerating relative to an inertial frame.

Key insight: Newton's laws only work correctly when acceleration is measured from an inertial reference frame.

Earth is considered an approximate inertial frame because:

• The sum of forces on an object at rest relative to Earth (e.g., a book on a table) is very close to zero, though not exactly zero • The deviation from a perfect inertial frame is negligible for most practical calculations • For routine affairs, "a⃗ = 0 if and only if F⃗ = 0" holds true with acceptable accuracy • A very precise measurement would show tiny discrepancies due to Earth's rotation and orbital motion

This approximation allows us to apply Newton's laws on Earth with high confidence for everyday physics problems.

A pseudo force:

• Is a fictitious force that appears in non-inertial reference frames
• Has magnitude m·a₀ where m is the mass of the object and a₀ is the acceleration of the reference frame
• Acts in the direction opposite to the acceleration of the reference frame
• Is necessary to make Newton's laws work in non-inertial frames

The pendulum in an accelerating car demonstrates this by:

• Appearing to deflect at angle θ where tanθ = a₀/g, despite no actual horizontal force acting on it
• Requiring a pseudo force ma₀ in calculations to explain why the pendulum is in equilibrium at this angle
• Showing that observers in the car must include this pseudo force to correctly analyze the system
• Illustrating that pseudo forces are proportional to the mass of the object, just like gravitational forces

The self-adjusting nature of frictional forces enables coordinated movement through:

1. Automatic equilibration of accelerations:

   • Forward friction f on horse adjusts to match needed propulsion
   • Backward friction f′ on cart adjusts based on rolling resistance

2. Force balancing mechanism:

   • These forces self-adjust such that (F₁-f′)/Mₖ = (f-F₂)/Mₕ
   • This equality ensures the horse and cart maintain the same acceleration

3. Practical implications:

   • Without this self-adjustment, the connection between horse and cart would either break or cause one to drag the other
   • The friction forces respond dynamically to changes in applied muscle force
   • This represents a natural feedback system where forces reach equilibrium values

This coordinated adjustment is why horse and cart move together as a system despite having different masses and experiencing different force components.

The tensions in the strings are:

$T₁ = \frac{mg \cos\beta}{\sin(\alpha+\beta)}$

$T₂ = \frac{mg \cos\alpha}{\sin(\alpha+\beta)}$

These expressions are derived by:

• Setting up force equilibrium equations (horizontal and vertical components)
• Horizontal: $T₁\cos\alpha - T₂\cos\beta = 0$
• Vertical: $T₁\sin\alpha + T₂\sin\beta - mg = 0$
• Solving these equations simultaneously for T₁ and T₂

Note: The denominator $\sin(\alpha+\beta)$ shows that as the strings approach the same line ($\alpha+\beta \rightarrow 180°$), the tensions approach infinity.

Systematic approach for force equilibrium problems:

1. Identify the system: Clearly define what object or collection of objects you're analyzing

2. Draw a free-body diagram:

   • Include ALL forces acting on the system
   • Use arrows to indicate direction and relative magnitude
   • Label each force with its symbol and source (e.g., T₁ by string 1)

3. Establish coordinate system:

   • Choose axes that simplify the problem (often along/perpendicular to inclines)
   • Consistently decompose forces along these axes

4. Apply equilibrium conditions:

   • Sum of forces equals zero: ΣF = 0
   • Write separate equations for each axis: ΣFₓ = 0, ΣFᵧ = 0

5. Solve the resulting equations:

   • Use simultaneous equations methods for multiple unknowns
   • Substitute known values last to avoid calculation errors

Example: For a suspended mass, decompose gravity plus tension forces along horizontal and vertical axes, set each sum to zero, then solve for unknown tensions.

Resolving forces into components is critical because:

1. Vector addition requirement: Forces are vectors and must be added vectorially, not arithmetically

2. Equilibrium conditions: For a body in equilibrium, the vector sum must equal zero, which is easiest to verify by component

3. Non-aligned forces: When forces act at different angles, direct summation is impossible without component analysis

4. Simplified equations: Component analysis converts a complex vector problem into simpler scalar equations

5. Independent equations: Each component direction produces a separate equation, providing enough relations to solve for multiple unknowns

Example process:

• For a mass suspended by strings at angles α and β:
  • Horizontal: T₁cosα - T₂cosβ = 0
  • Vertical: T₁sinα + T₂sinβ - mg = 0
• These independent equations allow solving for both T₁ and T₂

Without component analysis, equilibrium problems with non-collinear forces would be effectively unsolvable.

For the system to remain at rest:

1. The angle θ must satisfy: sin θ = m₂/m₁
2. This is derived from the equilibrium condition where:
   • The hanging mass exerts tension T = m₂g
   • This tension must equal the component of m₁'s weight parallel to the incline: m₁g sin θ
   • Setting these equal: m₂g = m₁g sin θ

Therefore, θ = sin⁻¹(m₂/m₁)

Note: This only works when m₂ < m₁, otherwise the system will accelerate.

Physical principles relating tension to string angles:

1. Vector nature of forces:

   • Tensions act along the direction of the strings
   • Only the vertical components support the weight
   • Horizontal components must balance each other

2. Mechanical advantage principle:

   • As a string approaches horizontal (angle → 0°), the tension approaches infinity
   • As a string approaches vertical (angle → 90°), the tension approaches its minimum value
   • For a vertical string, tension equals the portion of weight it supports

3. Mathematical relationship:

   • Tension in a string = (Portion of weight supported) ÷ (sine of angle with horizontal)
   • Example: If a single string at 30° supports a 10N weight, the tension would be 10N ÷ sin(30°) = 20N

4. Practical implication:

   • Shallow angles require significantly greater tensions to support the same weight
   • This explains why tightrope walkers use highly tensioned cables