Newton’s Laws of Motion and Dynamics Study Guide

Foundations of Dynamics and Newton’s Historical Context

• Sir Isaac Newton's Legacy: The Englishman Sir Isaac Newton published Philosphiae Naturalis Principia Mathematica (The Mathematical Principles of Natural Philosophy) in 1687. Today, this work is commonly referred to as The Principia.

• The Birth of Modern Physics: The Principia marked the transition of physics into a formal scientific discipline. The three laws contained within form the foundation of Dynamics.

• Kinematics vs. Dynamics:     * Kinematics: Focuses on the nature of how objects move.     * Dynamics: Focuses on the cause of motion (why objects move the way they do).

• Philosophical Stance: Newton famously noted, ‘‘Nature is pleased with simplicity. And nature is no dummy.’’

Introduction to Forces and Interactions

• Definition of Force: An interaction between two bodies characterized as a push or a pull. Forces are necessary for actions to occur.

• Examples of Force Interactions:     * Gravitational Force: The Earth exerting a downward pull on an object, such as an apple falling from a tree.     * Normal Force: The upward supporting force provided by a surface (like a floor) when an object stands on it.     * Frictional Force: The force exerted by a surface against an object sliding across it (e.g., a crate moving across a floor).

Newton’s First Law of Motion (The Law of Inertia)

• Definition: An object will continue in its current state of motion unless compelled to change by an impressed force.

• Condition of Equilibrium: Unless an unbalanced force acts on an object, its velocity will not change:     * If the object is at rest, it remains at rest.     * If the object is in motion, it continues at a constant speed in a straight line.

• Inertia: The natural resistance of objects to changes in their state of motion. The First Law is frequently called the Law of Inertia.

• Velocity Constancy: Constant velocity implies zero acceleration ($a = 0$), which in turn implies that the net force ($F_{net}$) is zero.

Newton’s Second Law of Motion

• Definition: Predicts the behavior of an object when an unbalanced force acts upon it. The object's velocity will change, meaning it will accelerate.

• Proportionality: Acceleration ($a$) is directly proportional to the strength of the net force ($F_{net}$) and inversely proportional to the object's mass ($m$).

• The Fundamental Equation of Mechanics:     * $a = rac{F}{m}$     * $F_{net} = ma$     * $\Sigma F = ma$

• Units: Force is measured in Newtons ($N$).     * $1\,N = 1\,kg \cdot m/s^2$     * A medium-size apple weighs approximately $1\,N$.

• Mass and Inertia:     * Mass ($m$) is a proxy for the inertia inherent in an object.     * Mass is measured in kilograms ($kg$).     * Comparison: $1\,kg$ of mass weighs approximately $2.2\,pounds$.     * If a force is applied to a $2\,kg$ object, a $1\,kg$ object would experience twice the change in velocity from that same force.

• Vector Nature: Forces have both magnitude and direction. The direction of acceleration ($a$) always matches the direction of the net force ($F_{net}$).

• Net Force (Resultant Force): The vector sum of all individual forces acting on an object simultaneously.

Newton’s Third Law of Motion

• Definition: Commonly remembered as ‘‘to every action, there is an equal, but opposite, reaction.’’

• Interaction Pairs: If Object 1 exerts a force on Object 2 ($F_{1\text{-on-}2}$), then Object 2 exerts a force back on Object 1 ($F_{2\text{-on-}1}$). These forces are equal in strength but opposite in direction.

• The Action/Reaction Pair: Symbolized as $F_{1\text{-on-}2} = -F_{2\text{-on-}1}$.

Concepts of Weight and Mass

• Distinction: In physics, mass and weight are not interchangeable.     * Mass: A measure of the quantity of matter in an object; it does not change based on location.     * Weight: The gravitational force exerted on an object by the Earth or another celestial body; it changes depending on the gravitational field of the location (e.g., an object weighs less on the Moon than on Earth).

• Weight Calculation: Weight is a force, calculated using Newton's Second Law where acceleration is gravitational acceleration ($g$):     * $F_w = mg$     * $F_g = mg$     * In the provided examples, the value of $g$ is approximated as $10\,m/s^2$.

• Unit Difference: Mass is in kilograms ($kg$); Weight is in Newtons ($N$).

The Normal Force

• Definition: A contact force exerted by a surface on an object. The word ‘‘normal’’ specifically means perpendicular.

• Direction: The normal force ($F_N$ or $N$) acts perpendicular to the surface of contact.

• Function: It acts as the supporting force that prevents objects from falling through floors or tables.

• Calculation: There is no universal formula for the normal force. It must be determined by analyzing all forces acting on the object and applying Newton's laws (e.g., on a flat table with no other vertical forces, $F_N = mg$).

Problem-Solving Strategy: The Free-Body Diagram (FBD)

1. Visualize and Sketch: Create a free-body diagram to represent the scenario.     * Use a dot to represent the object.     * Draw arrows pointing away from the dot to represent all forces acting on the object (not velocity).     * Ropes: Can only pull.     * Surfaces: Exert normal force (perpendicular) and friction (parallel).     * Gravity: Always points down (toward the center of the Earth).     * Arrow Length: Should reflect the relative magnitude of the forces.

2. Define Coordinate System: Choose axes; break any forces not on the axes into $x$ and $y$ components.

3. Newton’s Second Law: Write out the equations for each axis:     * $\Sigma F_x = ma_x$     * $\Sigma F_y = ma_y$

4. Math: Solve the resulting equations.

Frictional Forces

• Definition: The component of the contact force parallel to the surface.

• Origin: Arises from electrical/electrostatic interactions between the atoms of the object and the surface.

• Static Friction ($f_s$):     * Occurs when there is no relative motion (sliding) between the object and the surface.     * Results from electrostatic bonds formed while the object is at rest.     * Variable limit: Static friction varies to counteract applied force up to a maximum value.     * Equation: $F_{static\,friction, max} = \mu_s F_N$.

• Kinetic Friction ($f_k$):     * Occurs when there is relative motion (sliding).     * Bonds cannot form fast enough during motion, making kinetic friction generally lower than static friction.     * Equation: $F_{kinetic\,friction} = \mu_k F_N$.

• Coefficient of Friction ($\mu$):     * A unitless number representing the nature of the two surfaces in contact.     * Higher $\mu$ implies stronger friction (e.g., rubber on wood is $0.7$, while rubber on ice is $0.1$).     * For any given pair of surfaces: \mu_k < \mu_s.

• Direction: Kinetic friction is always opposite the direction of sliding. Static friction is usually opposite the direction of intended motion.

Pulleys and Tension

• Definition of Tension: The pulling force in a cord, rope, or string. Its direction is opposite the force creating the tension.

• Pulleys: Simple machines that change the direction of the tension force. They can multiply force based on the number of strings pulling on the object.

• System Acceleration: If the net force in a pulley system is not zero, the objects in the system are accelerating.

Inclined Planes (Ramps)

• Coordinate Rotation: To simplify math, use a rotated coordinate system where the x-axis is parallel to the ramp and the y-axis is perpendicular to it.     * Normal force ($F_N$) acts only on the y-axis (perpendicular).     * Friction ($F_f$) and Acceleration ($a$) act only on the x-axis (parallel).     * Gravity ($F_w = mg$) must be broken into components.

• Components of Gravity on a Ramp:     * Parallel component: $mg \sin(\theta)$. This is the force driving the object down the incline.     * Perpendicular component: $mg \cos(\theta)$. On a simple ramp, this is equal in magnitude to the normal force ($F_N = mg \cos(\theta)$).

Educational Sidebars and Contextual Notes

• Big Idea #1: Per College Board AP Physics 1 curriculum. Note that concepts were reorganized into 10 units starting in the 2019–2020 school year.

• Big Idea #3: Related to Free-Body Diagrams and force interactions.

• Common Pitfalls:     * Do not draw velocity vectors on a Free-Body Diagram.     * There is no such thing as a ‘‘force of inertia.’’ Inertia is a property of mass, not a force.     * Normal force is not always equal to weight; it depends on the angle of the surface and other applied forces.

Comprehensive Worked Examples

• Example 1: Net force for a $5,000\,kg$ object at constant velocity of $7,500\,m/s$.     * Solution: Constant velocity means $a = 0$. Per Newton's First Law, $F_{net} = 0$. Force is required to counteract friction, but the net force is zero.

• Example 2: Force for a $2\,kg$ object with $a = 4\,m/s^2$.     * Solution: $F_{net} = ma = (2\,kg)(4\,m/s^2) = 8\,N$.

• Example 3: Object with $8\,N$ force left and $20\,N$ force right ($mass = 4\,kg$).     * Solution: $F_{net} = 20\,N - 8\,N = 12\,N$ to the right. $a = \frac{12\,N}{4\,kg} = 3\,m/s^2$ to the right.

• Example 4: Mass of an object weighing $500\,N$.     * Solution: $m = \frac{F_w}{g} = \frac{500\,N}{10\,m/s^2} = 50\,kg$.

• Example 5: Mass of a $150\,lb$ person ($1\,lb = 4.45\,N$).     * Solution: Weight in Newtons = $(150)(4.45) = 667.5\,N$. $m = \frac{667.5\,N}{10\,m/s^2} = 66.75\,kg$.

• Example 6 & 7: $2\,kg$ book at rest on a table.     * Solution: $a = 0 \implies F_{net} = 0$. Upward supporting force (Normal Force) must equal weight: $(2\,kg)(10\,m/s^2) = 20\,N$. $F_N = 20\,N$.

• Example 9: $6\,kg$ paint can pulled up with $a = 1\,m/s^2$.     * Solution: $F_T - F_w = ma \implies F_T = m(g + a) = 6(10 + 1) = 66\,N$.

• Example 10: $6\,kg$ paint can pulled up at constant velocity ($v = 1\,m/s$).     * Solution: constant velocity $\implies a = 0 \implies F_T = F_w = (6)(10) = 60\,N$.

• Example 11: Lifting a $50\,N$ object with $a = 10\,m/s^2$.     * Solution: $F_w = 50\,N$, so $m = 5\,kg$. $F_T = F_w + ma = 50 + (5)(10) = 100\,N$.

• Example 12: $20\,kg$ crate, $\mu_k = 0.3$, applied force $90\,N$.     * Solution (a): $F_N = mg = 200\,N$. $F_k = (0.3)(200) = 60\,N$.     * Solution (b): $F_{net} = 90 - 60 = 30\,N$. $a = \frac{30}{20} = 1.5\,m/s^2$.

• Example 13: $100\,kg$ crate, $\mu_s = 0.4$, applied force $250\,N$.     * Solution: $F_{s, max} = (0.4)(1,000\,N) = 400\,N$. Since 250\,N < 400\,N, the crate stays at rest and the actual static friction force is $250\,N$.

• Example 14: Frictionless pulley system (tabletop) with blocks $m$ and $M$.     * Solution: For $m$: $F_T = ma$. For $M$: $Mg - F_T = Ma$. Adding equations: $Mg = (m+M)a \implies a = \frac{Mg}{m+M}$.

• Example 15: Pulley system with friction ($m = 2\,kg, M = 10\,kg, \mu_k = 0.5$).     * Solution: $F_f = \mu mg = (0.5)(2)(10) = 10\,N$. $F_{net} = Mg - F_f = (10)(10) - 10 = 90\,N$. Total mass = $12\,kg$. $a = \frac{90}{12} = 7.5\,m/s^2$ .

• Example 17: Frictionless $30^{\circ}$ incline.     * Solution: $a = g \sin(30^{\circ}) = 10(0.5) = 5\,m/s^2$.

• Example 18: $30^{\circ}$ incline with $\mu_k = 0.3$.     * Solution: $a = g(\sin(\theta) - \mu \cos(\theta)) = 10(\sin(30^{\circ}) - 0.3 \cos(30^{\circ})) \approx 2.4\,m/s^2$.