Notes on Newton's Laws: First, Second, and Third Law with Examples

Newton's Laws: Detailed Notes from the Transcript

  • Newton's First Law (Law of Inertia)

    • Core idea: An object not acted on by any external force moves with constant velocity along a straight line (rectilinear motion). In symbols, if the external force is zero, the acceleration is zero: F⃗ext=0⇒a⃗=0.\vec{F}_{ext}=0 \Rightarrow \vec{a}=0. When a nonzero external force acts, motion changes accordingly.
    • Everyday intuition vs. reasoning: The idea that a wrench would keep moving in a straight line forever if far from gravity is an abstraction. In the real world, external forces (gravity, friction, etc.) are typically present, so motion changes. Yet Newton reasoned the law to be a fundamental description of motion in the absence of external forces.
    • Illustrative anecdote: A documentary about Titan rocket protocol change (from tightening with a wrench to using a socket). A worker forgot the socket and chose to use the wrench, tightening at a constant speed. This story is used to motivate thinking about motion under forces and the limits of everyday observations.

  • Newton's Second Law

    • Formal relationship: The net external force on an object equals its mass times its acceleration. In vector form: F⃗net=m a⃗.\vec{F}_{net}=m\,\vec{a}.
    • Frictionless surface example: A block on a floor with a net push causes acceleration. The block accelerates in the direction of the net force; the mass resists acceleration (larger mass -> smaller acceleration for a given force).
    • Key concept: Mass is the resistance to acceleration; heavier objects require more force to achieve the same acceleration.
    • Single-force form: If only one force acts, F⃗=ma⃗.\vec{F}=m\vec{a}. If multiple forces act, use the net force: F⃗<em>net=∑</em>iF⃗i.\vec{F}<em>{net}=\sum</em>i \vec{F}_i.
    • Algebraic intuition: With a fixed force, increasing mass reduces acceleration: a⃗=F⃗netm.\vec{a}=\frac{\vec{F}_{net}}{m}. As m grows large, acceleration approaches zero but is not necessarily zero unless force is zero.
    • Free-body diagram (FBD) utility: To apply Newton's 2nd law, draw an FBD for the object: represent the object as a dot and sketch all forces acting on it (gravity, normal force, friction, applied forces).
    • Weight and normal force example:
    • Gravity force: F⃗g=mg⃗,\vec{F}_g=m\vec{g}, acting downward (where g ≈ 9.81 m/s^2).
    • Normal force: N⃗,\vec{N}, perpendicular to the contact surface, balancing perpendicular components when there is no vertical acceleration.
    • Example numbers from the transcript: A mass is acted on by horizontal forces (e.g., 50 N to the right, 20 N to the left) on a frictionless surface. The net force is F⃗<em>net=50 N−20 N=30 N,\vec{F}<em>{net}=50\,\mathrm{N}-20\,\mathrm{N}=30\,\mathrm{N}, and the resulting acceleration is a⃗=F⃗</em>netm.\vec{a}=\dfrac{\vec{F}</em>{net}}{m}. If the mass is 4 kg, then a⃗=304=7.5 m/s2.\vec{a}=\dfrac{30}{4}=7.5\ \text{m/s}^2.
    • Worked scenario: If instead the problem gives acceleration directly (e.g., a = 2 m/s^2) and mass m = 4 kg, the net force would be F⃗net=m a⃗=4 kg⋅2 <br/>m/s2=8 N.\vec{F}_{net}=m\,\vec{a}=4\,\mathrm{kg} \cdot 2\,<br /> \mathrm{m/s^2}=8\ \mathrm{N}. Then you can use FBDs to determine unknown forces that would produce that net acceleration.
    • Practical note: When there are multiple forces, you must sum them vectorially to get the net force; the acceleration is then in the direction of that net force.

  • Newton's Third Law (Action-Reaction)

    • Core idea: Forces come in equal and opposite pairs acting on two interacting bodies. If one object A pushes on B with a force (\vec{F}{A\,on\,B}), then B pushes on A with force (\vec{F}{B\,on\,A}=-\vec{F}_{A\,on\,B}).
    • Boxing/pushing analogy: A boxer punches a bag that has substantial mass; the bag can exert a large reaction force on the boxer, enabling a large push. If you punch a light object (e.g., Kleenex), the reaction force is small, so you can't push hard.
    • Ball vs. window example (contact forces): When a ball hits a plate glass window, the force the ball exerts on the window and the force the window exerts on the ball are equal and opposite while in contact. The graph of force vs. time during impact would show the force rising as contact is made and then dropping when the window breaks, with the forces equal and opposite during the contact interval.
    • Non-contact example: Apple and Earth under gravity. The Earth exerts a downward force on the apple, and the apple exerts an equal and opposite force on the Earth. Since the Earth is ~massive, its acceleration is negligible, so we don’t observe motion of the Earth, but the forces are still equal and opposite.
    • Important nuance: The equal-and-opposite forces act on different bodies, so they do not cancel in the same net-force calculation for a single body. For the cart example discussed in the transcript, the force that the other object exerts on the cart must be included in the net force on the cart; summing the action-reaction pair forces as if they acted on the same body would incorrectly suggest zero net force.

  • Free-Body Diagrams (FBDs) and Net Force

    • Procedure:
    • Represent the object as a dot.
    • Draw all forces acting on the object with arrows at the point of contact or direction of action (vector quantities).
    • Common forces on a block on a surface:
    • Gravity: F⃗g=mg⃗\vec{F}_g=m\vec{g} downward.
    • Normal force: N⃗\vec{N} exerted by the surface, perpendicular to the surface (often upward in vertical balance).
    • Friction: F⃗f\vec{F}_f (if present), opposing motion along the surface.
    • Applied forces: any external forces you apply.
    • On a frictionless horizontal surface with a single horizontal net force, the vertical forces balance (N = mg) and the horizontal net force determines horizontal acceleration via F⃗<em>net=ma⃗.\vec{F}<em>{net}=m\vec{a}. For a specific numerical example, with horizontal forces such as 50 N to the right and 20 N to the left, the net horizontal force is F⃗</em>net=50 N−20 N=30 N,\vec{F}</em>{net}=50\,\mathrm{N}-20\,\mathrm{N}=30\,\mathrm{N}, leading to the horizontal acceleration a⃗=F⃗netm.\vec{a}=\dfrac{\vec{F}_{net}}{m}.
    • Key teaching point: Always identify which forces act on the particular object you are analyzing; do not double-count action-reaction pairs on the same body.

  • Mass, Weight, and Units

    • Mass (intrinsic property): Resistance to acceleration. It determines how much the velocity of an object changes under a given force.
    • Weight: W=m gW=m\,g, the force due to gravity acting on a mass near a planet's surface.
    • Units:
    • Force: N\mathrm{N} (newton), where 1 N=1 kg m s−2.1\ \mathrm{N}=1\ \mathrm{kg\, m\, s^{-2}}.
    • Mass: measured in kilograms (kg).
    • Acceleration: measured in meters per second squared (m/s^2).
    • Relationship to acceleration: a⃗=F⃗netm\vec{a}=\dfrac{\vec{F}_{net}}{m}, which arises from dimensional analysis: N=kg m/s2.\mathrm{N}=\mathrm{kg}\, \mathrm{m/s^2}. In words: applying a force to a 4 kg mass with a net force of 8 N produces an acceleration of a=8 N4 kg=2 m/s2.a=\dfrac{8\ \mathrm{N}}{4\ \mathrm{kg}}=2\ \mathrm{m/s^2}.

  • Velocity and Acceleration in Time (Graphs and Kinematics)

    • If acceleration is constant, velocity changes linearly with time: v(t)=v0+at.v(t)=v_0+a t. If the initial velocity is zero, then v(t)=at.v(t)=a t.
    • This connects to net force via Newton's second law: a is determined by the net force and mass; if you know F_net and m, you know how velocity evolves over time.

  • Practical and Conceptual Connections

    • Real-world relevance: Newton's laws underpin trajectories, safety design, and motion analysis in engineering, sports, and everyday tasks.
    • Foundational principle: Free-body diagrams are essential tools to identify net forces and apply F⃗net=ma⃗.\vec{F}_{net}=m\vec{a}.
    • Limitations and assumptions discussed:
    • Frictionless vs. real surfaces: Many examples assume no friction to isolate the effect of the net horizontal force; real scenarios require including friction and other resistive forces.
    • Nonzero external forces: The first law presumes no external force; in practice there are always some external influences unless a perfect vacuum frame is considered.

  • Summary of Key Equations (LaTeX)

    • First Law concept (in absence of external force): F⃗ext=0⇒a⃗=0.\vec{F}_{ext}=0 \Rightarrow \vec{a}=0. For motion in a straight line with nonzero external forces, the change in velocity is governed by those forces.
    • Second Law (net force): F⃗<em>net=ma⃗\vec{F}<em>{net}=m\vec{a} and for multiple forces: F⃗</em>net=∑<em>iF⃗</em>i.\vec{F}</em>{net}=\sum<em>i \vec{F}</em>i.
    • Third Law (action-reaction): F⃗<em>A on B=−F⃗</em>B on A.\vec{F}<em>{A\,on\,B}= -\vec{F}</em>{B\,on\,A}.
    • Weight and Normal Force: F⃗g=mg⃗,N⃗ is normal to the surface.\vec{F}_g=m\vec{g}, \quad \vec{N} \text{ is normal to the surface}.
    • Velocity with constant acceleration: v(t)=v0+at.v(t)=v_0+a t.
    • Newton's second law units consistency: 1 N=1 kg m s−2.1\ \mathrm{N}=1\ \mathrm{kg\, m\, s^{-2}}.

  • Example Takeaways for Exam Preparation

    • Be able to identify and compute net forces on a single object using a free-body diagram.
    • Practice translating a verbal scenario into a mathematical model: choose coordinate directions, sum forces, apply F⃗net=ma⃗\vec{F}_{net}=m\vec{a}, and solve for the unknowns.
    • Distinguish between forces acting on one object and the equal-and-opposite forces acting on another object in interaction scenarios (action-reaction pairs).
    • Use the constant-acceleration model to relate force, mass, and motion over time for simple problems, including both horizontal and vertical directions.