Lecture 4: Newton’s Second Law

Introduction to Newton’s Second Law

• Newton’s second law of motion describes the fundamental relationship between force, mass, and acceleration.

• This law serves as a bridge for future discussions involving momentum, energy, and more advanced mechanics.

• The primary focus includes defining force, articulating Newton's second law, and distinguishing between mass and weight.

Force: Definition, Symbol, and Units

• Simply put, a force is a push or pull that causes an object to change its motion.

• Symbol: The capital letter $F$ is used to represent force.

• Vector Nature: Force is a vector quantity, meaning it has both magnitude and direction. In text, it is denoted by bold font or an arrow over the letter ($\vec{F}$).

• Units: The standard unit for force is the Newton ($N$).

• Dimensional Analysis: One Newton is equal to one kilogram times a meter divided by a second-squared:   $1\,N = 1\,\frac{kg \times m}{s^2}$

Detailed Taxonomy of Forces

Gravity

• Gravity is a constant downward pull toward the center of the Earth caused by the Earth’s large mass.

• It is responsible for the motion of objects dropping or rolling down hills and ramps.

• Weight is a direct result of the gravitational force acting on an object's mass.

Support Force (Normal Force)

• Occurs whenever an object is in contact with a surface (floor, wall, etc.).

• The support force prevents gravity from accelerating an object toward the Earth's center when it is stationary on a surface.

• It is often called the "Normal Force" because it always acts perpendicular ($\perp$) to the supporting surface.

• For a flat surface, the normal force points directly up; if the surface is tilted, the force points perpendicular to that tilt.

Friction

• Friction is the force experienced in daily activities resisting motion between surfaces.

• Static Friction: The force required to initiate motion for an object at rest. It typically takes more effort to start an object moving than to keep it moving due to the higher resistance of static friction.

• Kinetic Friction: The force that occurs between surfaces as an object slides. The magnitude depends on the materials in contact (e.g., steel on ice has low friction; wood on steel has much higher friction).

• Influence of Support Force: The magnitude of kinetic friction is proportional to the support force between the object and the surface.

Air Resistance (Drag)

• This force occurs when an object travels through a fluid, such as air, acting in the opposite direction of the motion.

• It is proportional to the speed of the object: higher speeds result in higher air resistance.

• Form Factor: Resistance depends on the shape of the object. A flat sheet of paper experiences more drag and falls slower than a crumpled sheet because its larger form factor causes more air to drag against it.

• Aerodynamics: The study of shapes designed to minimize drag, crucial for aircraft to reduce fuel costs.

• Utility: Parachutes use air resistance to decrease terminal speed and make falls survivable.

• Terminal Speed: As an object falls, air resistance increases with speed until it equals the force of gravity. At this point, the object moves at a constant speed called terminal speed, achieving dynamic equilibrium. Terminal speed varies based on size, shape, and density.

Applied and Tension Forces

• Applied Force: A push or pull applied to an object by an outside influence.

• Tension: The pulling force present in strings, cables, or ropes holding something up. The direction is always along the line of the string/cable. It is found in pendulums, pulley systems, and truss bridges.

Elasticity

• The ability of objects to deform under force and return to their original configuration once the force is removed.

• Springs are the primary example; they restore themselves provided the applied force was not excessive.

Electromagnetic Forces

• Describes interactions between charged objects and magnetic materials.

• This accounts for electronic gadgets and refrigerator magnets.

• Most macroscopic forces, such as friction and support forces, are fundamentally electromagnetic (e.g., electron repulsion between atoms in contact).

Nuclear forces: Strong and Weak

• These forces operate within the nucleus of an atom.

• Strong Force: Acts over very small scales to keep the nucleus together by overriding the electromagnetic repulsion between positively charged protons. It is vital for understanding nuclear fusion, fission, and radioactive decay.

• Weak Force: Responsible for radioactive decay and nuclear processes. It can change one type of quark into another (e.g., turning a proton into a neutron).

The Four Fundamental Forces

All forces in the universe can be categorized into four fundamental forces. Ranked from strongest to weakest, they are:

1. The Strong Force (Strongest)

2. Electromagnetism

3. The Weak Force

4. Gravity (Weakest)

Newton’s Second Law: Force, Mass, and Acceleration

• While acceleration is the rate of change of velocity, Newton's second law defines its cause through force and mass.

• Equation: The net force on an object is equal to its mass times its acceleration.   $F_{net} = m \times a$

• Vector Relationship: Acceleration is a vector that points in the same direction as the net force.

• Mechanical Equilibrium: If the net force is zero ($F_{net} = 0$), then the acceleration is zero ($a = 0$).

Quantitative Scenarios and Examples

Scenario 1: Cart on a Ramp

• Given Mass ($m$): $0.35\,kg$

• Given Acceleration ($a$): $0.9\,m/s^2$

• Calculation:   $F_{net} = 0.35\,kg \times 0.9\,m/s^2 = 0.315\,N$

• The net force of $0.315\,N$ is the sum of gravity, support force, friction, and air drag, and it points down the ramp parallel to the motion.

Scenario 2: Static Equilibrium (Pushing a Box)

• Problem: You push a heavy box with $25\,N$ of force, but it does not move.

• Analysis: Because it is at rest, it is in static equilibrium and $F_{net} = 0$.

• Formula: $F_{net} = F_{applied} + F_{friction} = 0$

• Plugging in values: $25\,N + F_{friction} = 0 \implies F_{friction} = -25\,N$

• Lesson: Friction and applied force have equal magnitudes but opposite directions. If friction were greater than the push, the box would accelerate toward you, which is physically impossible.

Two-Dimensional Force Analysis

• Forces in 2D can be analyzed using a force table by resolving vectors into horizontal and vertical components.

• Demo Setup:

  • $55\,g$ ($0.55\,N$) at $0^{\circ}$

  • $90\,g$ ($0.9\,N$) at $110^{\circ}$

  • Horizontal Net Force: $0.55\,N + (-0.31\,N) = 0.24\,N$

  • Vertical Net Force: $0.85\,N$

• Equilibrium Requirement: To achieve equilibrium, a force must be added that cancels these components. This calculates to a force with components of $(-0.24, -0.85)\,N$, which corresponds to placing $88\,g$ at $254^{\circ}$.

• Graphical Method: Vectors can be added "tail to tip" on graph paper or using online simulators, then solved using the Pythagorean theorem for magnitude:   $c = \sqrt{a^2 + b^2}$

Mass vs. Weight and Gravity

• Mass ($m$): The amount of "stuff" (atoms/molecules) in an object. It represents inertia (resistance to motion) and is constant regardless of location.

• Weight ($W$): A force resulting from gravity.   $W = m \times g$

• Gravitational Acceleration ($g$): On Earth, $g = -9.8\,m/s^2$.

• Variable Weights by Location:

  • Moon: Gravity is $1/6$ of Earth’s.

  • Jupiter: Gravity is $2.5 \times$ higher than Earth’s.

  • Sun: Gravity is nearly $30 \times$ Earth's gravity.

Weightlessness and Inertia in Space

• Weightlessness: Astronauts in orbit appear weightless because they lack a support force, though they are still under the influence of Earth's gravity (which maintains the orbit).

• Persistence of Inertia: Even in microgravity, mass and inertia still affect movement.

• Astronaut Insights (Les Padilla, NASA): Crew members must move very slowly along the space station. Getting a $300\,lb$ mass moving in microgravity is dangerous because while it may be weightless, its inertia makes it difficult to stop, resulting in wasted energy or potential injury.

Historical Demonstrations

• Apollo 15 (1971): Astronaut Dave Scott dropped a hammer and a feather simultaneously on the moon.

• Result: Because the moon has no atmosphere (and thus no air resistance), the feather and hammer hit the surface at exactly the same time, proving that gravity accelerates all objects equally in a vacuum.