Newton’s Laws of Motion

A reference frame moving relative to Earth is inertial if and only if:

1. The frame moves with constant velocity (uniform motion) relative to Earth
2. There is no rotation of the frame

Examples:

• A train moving at constant speed on straight tracks
• An airplane flying at constant speed and altitude
• A car traveling at constant velocity on a straight road

Mathematical relationship: If S is an inertial frame and S' moves uniformly relative to S, then:

• a⃗ₚ,ₛ = a⃗ₚ,ₛ' (accelerations measured in both frames are identical)
• Therefore S' is also inertial

This is why Newton's laws work equally well on a smoothly cruising airplane as they do on the ground.

Systematic algorithm for applying Newton's laws:

1. Decide the System: • Identify what object(s) to analyze • Ensure all parts have identical acceleration • May be a single particle, block, or multiple connected objects

2. Identify All Forces: • List ONLY forces acting ON the system • Include only forces from objects EXTERNAL to the system • Categorize by type (gravitational, electromagnetic, etc.)

3. Create Free Body Diagram: • Represent system as a point • Draw vectors for all forces with common origin • Indicate magnitudes and directions

4. Choose Axes and Write Equations: • Orient X-axis along expected acceleration direction • Write ΣF₍ₓ₎ = ma₍ₓ₎ • Write ΣF₍y₎ = ma₍y₎ (often = 0 for equilibrium in Y) • Add kinematic relationships between objects if needed

5. Solve the resulting equations mathematically

Inertia vs. Mass:

Inertia:

• The property of an object that resists changes to its state of motion
• Manifests as the object's "unwillingness" to accelerate when forces are applied
• A qualitative property describing resistance to motion changes

Mass:

• The quantitative measure of inertia
• Scalar quantity defined by m = F/a (from Newton's Second Law)
• Determines the acceleration produced by a given force

Relationship:

• Larger mass = greater inertia = smaller acceleration for a given force
• Mass provides the numerical value for the amount of inertia

Key distinction:

• Inertia is the physical property/concept
• Mass is the mathematical measure of this property
• They're inseparable but conceptually distinct

Example: Two objects might both resist acceleration (show inertia), but the one with greater mass exhibits greater resistance for a given force.

To determine which components can be treated as a single system:

1. Acceleration criterion:

   • Components must share identical acceleration vectors (same magnitude AND direction)
   • Objects on the same horizontal or vertical line connected by inextensible strings typically move together
   • Objects moving perpendicular to each other must be treated separately

2. Connection analysis:

   • Rigid connections (blocks stacked without slipping) maintain same acceleration
   • String connections:
     • If string length between components remains constant, they move equal distances
     • If a string passes over a pulley, direction changes but displacement magnitudes remain equal
     • If string slides over another component, independent analysis is required

3. Practical application:

   • Blocks A and B stacked together without slipping: single system
   • Blocks connected by a taut horizontal string: single system
   • Blocks connected by a string passing over pulleys with perpendicular motion: separate systems
   • A pulley that slides along a string: separate system from other components

Example: In a person pulling a block via a pulley system, the block, string, and any intermediate blocks moving in the same direction can be treated as one system, while elements moving vertically when others move horizontally must be analyzed separately.

Pseudo forces (also called inertial forces) are fictional forces that must be introduced when analyzing motion from a non-inertial reference frame while still using Newton's laws in their standard form. Key characteristics:

• Mathematical form: F_pseudo = -ma₀ (where m is the object's mass and a₀ is the acceleration of the reference frame)
• Direction: Always opposite to the acceleration of the reference frame
• Purpose: Allows F=ma to work in non-inertial frames
• Not a real force: No physical interaction creates this force
• Examples: The apparent force pushing you back in an accelerating car

These forces are necessary because Newton's laws are only valid in inertial reference frames. By adding pseudo forces, we can apply F=ma in accelerating reference frames while getting correct results.

The resolution involves considering all forces in the complete system:

1. The horse exerts force F₁ on the cart (forward)
2. The cart exerts equal force F₂ on the horse (backward)
3. Crucially, the horse exerts force on the ground, which responds with force P with forward component f
4. The road exerts force Q on the cart with possible backward component f′

For motion to occur:

• The forward component f must exceed F₂
• The net acceleration depends on: (f - F₂)/Mₕ for the horse and (F₁ - f′)/Mₖ for the cart

This demonstrates that considering only the action-reaction pair between horse and cart is incomplete; interaction with the ground is essential for forward acceleration.

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.

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.

As the angle between the strings approaches 180°:

• The tensions in both strings approach infinity
• Mathematically: if α + β → 180°, then sin(α+β) → 0, causing: $T₁ = \frac{mg\cos\beta}{\sin(\alpha+\beta)} \rightarrow \infty$ $T₂ = \frac{mg\cos\alpha}{\sin(\alpha+\beta)} \rightarrow \infty$

Physical explanation:

• When strings are nearly collinear, their ability to provide vertical support diminishes
• Most of their tension produces horizontal forces that cancel each other
• Very little vertical component remains to support the weight
• To maintain equilibrium, the tensions must increase dramatically
• This creates a mechanical disadvantage similar to trying to lift a weight with nearly horizontal ropes

Practical implications:

• This principle explains why tightropes must be very taut
• It's also why cables with minimal sag require extremely high tension

Defining Characteristics

Inertial Reference Frames:

• Newton's First Law applies directly (objects with F⃗ = 0 move at constant velocity)
• Newton's Second Law holds in its standard form (F⃗ = ma⃗)
• All inertial frames move at constant velocity relative to each other
• No preferred "absolute rest" frame exists among inertial frames

Non-inertial Reference Frames:

• Accelerating relative to inertial frames (rotating, decelerating, etc.)
• Objects appear to accelerate even when no real forces act on them
• Newton's laws appear to fail without modification

Mathematical Criteria & Transformations

1. Inertial Frame Test:

   • A frame S' moving with velocity v⃗ relative to inertial frame S is inertial if and only if dv⃗/dt = 0
   • Free particles move in straight lines at constant speeds
   • Acceleration relationship: a⃗ₚ,ₛ = a⃗ₚ,ₛ' across inertial frames

2. Transformation between frames:

   • Position: r⃗ₛ = r⃗ₛ' + v⃗t + r⃗₀
   • For non-inertial frames, must account for frame acceleration

Practical Implications

• In inertial frames: Apply Newton's laws directly
• In non-inertial frames: Must introduce fictitious/pseudo forces (−ma⃗₀)

Example: In a car accelerating forward, you feel "pushed" backward not from a real force but due to inertia as observed from the car's non-inertial frame.

Key test: Place an object at rest and observe if it remains at rest without external forces. If it doesn't, you're in a non-inertial frame.