Newton's Laws of Motion and Applications: Study Notes

Ch. 5: Newton's Laws of Motion

5.1: Force and Inertia

Understanding force and inertia is fundamental to Newtonian mechanics. A force is any push or pull that can cause an object to accelerate, while inertia is the property of an object to resist changes in its state of motion.

• Force: An outside influence on an object that causes it to undergo a change in its motion (an acceleration).

• Force causes acceleration, not motion itself.

• Inertia: The tendency of an object to maintain its state of motion. It is quantified by mass ($\text{inertia} = \text{mass}$).

Key Point: Force does NOT cause motion; it causes changes in motion (acceleration).

5.2: Newton's First Law of Motion (Law of Inertia)

Newton's First Law states that an object at rest remains at rest, and an object in motion remains in motion with constant velocity, unless acted upon by a net external force.

• Rest: No net force means no change in motion.

• Constant velocity: Motion continues unchanged unless a net force acts.

5.3: Newton's Second Law of Motion

This law quantifies the relationship between force, mass, and acceleration. The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

• Mathematical Form: $\vec{F}_{\text{net}} = m \vec{a}$

• Units: Newton (N), where $1\,\text{N} = 1\,\text{kg} \cdot \text{m}/\text{s}^2$

• Acceleration: $a = \frac{F}{m}$

Example: A golf ball (0.05 kg) is hit to a speed of 45 m/s in 1 ms. The force is calculated using $F = m a$ and $a = \frac{\Delta v}{\Delta t}$.

Example: A 1200 kg car decelerates from 20 m/s to rest in 8 s. The braking force is found using $F = m a$ and $a = \frac{\Delta v}{\Delta t}$.

5.3b: Free-Body Diagrams (FBDs)

Free-body diagrams are visual representations of all forces acting on an object. They are essential for analyzing physical situations.

• Examples:

  • Box in space (only applied force)

  • Rock in free fall (only gravity)

  • Falling rock with air resistance (gravity and drag)

  • Book on a table (gravity and normal force)

• Normal Force: The support force exerted by a surface, always perpendicular to the surface.

5.3c: Tension and Weight

Tension is the force transmitted through a string, rope, or cable when it is pulled tight by forces acting from opposite ends. Weight is the force of gravity acting on an object.

• Tension (T): Acts along the rope, string, etc. Unit: Newton (N).

• Weight (W): $F_g = W = m g$ (vector, points toward Earth's center).

Example: Calculating tension in an elevator cable or in wires holding a painting at an angle.

5.3d: Inclined Planes and Multiple Forces

When analyzing objects on inclines or with multiple forces, resolve forces into components and apply Newton's laws in each direction.

• Example: Block sliding down a frictionless incline: $a = g \sin \theta$

• Example: Block with forces in two perpendicular directions: Use vector addition to find net force and acceleration.

5.3e: Spring Force (Hooke's Law)

The force exerted by a spring is proportional to its displacement from equilibrium, described by Hooke's Law.

• Formula: $F_{\text{spr}} = -k \Delta x$

• Spring constant (k): Measures stiffness (units: N/m).

Example: A block attached to a spring stretches it by a certain amount; use $F = k \Delta x$ to solve for displacement or force.

5.4: Newton's Third Law of Motion

For every action, there is an equal and opposite reaction. Forces always occur in pairs, acting on different objects.

• Action-Reaction Pairs: If object A exerts a force on object B, then B exerts an equal and opposite force on A.

• Key Point: The two forces in a pair never act on the same object, so they do not cancel each other for a single object.

Example: Astronaut pushes a box in space; both experience equal and opposite forces.

Ch. 6: Applications of Newton's Laws

6.1: Friction

Friction is a force that opposes the relative motion of two surfaces in contact. There are two main types: kinetic and static friction.

• Kinetic Friction ($f_k$): Acts on moving objects. $f_k = \mu_k F_N$

• Static Friction ($f_s$): Acts on stationary objects, preventing motion up to a maximum value. $0 \leq f_s \leq \mu_s F_N$

• Coefficient of Friction ($\mu$): Dimensionless constant depending on the surfaces in contact.

• Normal Force ($F_N$): The perpendicular force exerted by a surface.

Example: Calculating the force needed to keep a box moving at constant velocity, or to overcome static friction and start motion.

6.1b: Static Friction

Static friction prevents motion until a threshold is reached. It is not constant, but increases up to a maximum value.

• Maximum Static Friction: $f_{s,\text{max}} = \mu_s F_N$

• Range: $0 \leq f_s \leq \mu_s F_N$

Example: Calculating the force needed to start moving a box, and the subsequent acceleration if kinetic friction applies.

6.2: Spring Force (Review)

Spring force is a restoring force, always directed toward equilibrium. The spring constant $k$ quantifies the stiffness of the spring.

• Formula: $F_{\text{spr}} = -k \Delta x$

• Units: $\text{N}/\text{m}$

Example: Calculating the force in a spring or the extension caused by a hanging mass.

6.3: Microscopic Models of Friction

Friction arises from microscopic interactions between surfaces. Two common models are:

• Stick-slip model: Surfaces stick due to microscopic contact, then slip when force overcomes static friction.

• Bonding model: Microscopic bonds form and break as surfaces move past each other.

Key Equations and Concepts Table

Concept	Equation	Description
Newton's 2nd Law	$\vec{F}_{\text{net}} = m \vec{a}$	Net force equals mass times acceleration
Weight	$F_g = m g$	Force of gravity on an object
Kinetic Friction	$f_k = \mu_k F_N$	Friction for moving objects
Static Friction	$0 \leq f_s \leq \mu_s F_N$	Friction for stationary objects
Spring Force	$F_{\text{spr}} = -k \Delta x$	Restoring force of a spring

Summary of Problem-Solving Steps

• Draw a free-body diagram for the object(s).

• Resolve all forces into components (especially on inclines).

• Apply Newton's laws in each direction (usually x and y axes).

• Solve for unknowns (acceleration, force, tension, etc.).

• Check units and physical reasonableness of your answer.

Additional info: Some diagrams and examples were inferred and expanded for clarity. The notes include both conceptual explanations and worked examples to illustrate the application of Newton's laws and friction.