Chapter 4: Forces and Newton's Laws of Motion

Chapter 4: Forces and Newton's Laws of Motion

4.1 The Concepts of Force and Mass

  • Definition of Force:

    • A force is defined as a push or a pull.

    • Types of Forces:

      • Contact Forces:

        • Arise from physical contact between objects.

      • Action-at-a-Distance Forces:

        • Do not require contact; include gravitational and electrical forces.

  • Force Representation:

    • Forces are represented by arrows.

    • The length of the arrow is proportional to the magnitude of the force.

      • Example: Difference between 15 N and 5 N arrows.

  • Mass Definition:

    • Mass is a measure of the amount of “stuff” contained in an object.

4.2 Newton’s First Law of Motion

  • Law Statement:

    • An object continues in a state of rest or in motion at a constant speed along a straight line unless compelled to change that state by a net force.

  • Net Force:

    • The net force is the vector sum of all of the forces acting on an object.

    • Unit of Force: Newton (N).

      • Example calculation of net forces:

        • Individual Forces: 10 N, 4 N, 6 N results in:
          extNetForce=10N−4N−6N=0Next{Net Force} = 10 N - 4 N - 6 N = 0 N

      • Further example providing multiple forces leads to a net force of 3 N, 4 N, and 5 N resulting in 64 N.

  • Inertia:

    • Defined as the natural tendency of an object to remain at rest or in motion at a constant speed along a straight line.

    • Mass quantitatively measures inertia.

      • SI Unit of Mass: kilogram (kg).

  • Inertial Reference Frame:

    • An inertial reference frame is one in which Newton’s law of inertia is valid.

    • Accelerating reference frames are considered non-inertial.

4.3 Newton’s Second Law of Motion

  • Mathematical Representation:

    • The net force is expressed mathematically as:
      ext{Net Force} ( extstyle{oldsymbol{ ext{F}}}) = extstyle{oldsymbol{m}} imes extstyle{oldsymbol{a}}

    • Here, the Greek letter sigma (oldsymbol{ ext{Σ}}) denotes summation of forces:
      extstyle{oldsymbol{F}}_{net} = extstyle{oldsymbol{ΣF}}

  • Newton’s Law Statement:

    • When a net external force acts on an object of mass mm, the acceleration aa that results is:

      • Directly proportional to the net force,

      • Inversely proportional to the mass,

      • Direction of acceleration is the same as the direction of the net force.
        extstyle{oldsymbol{F}}{net} = m imes a extstyle{oldsymbol{a}} = rac{ extstyle{oldsymbol{F}}{net}}{m}

  • SI Unit for Force (N):

    • N=rackgimesms2N = rac{kg imes m}{s^2}

    • A newton (N) combines these units, showing how force relates to mass and acceleration.

  • Table of Units for Mass, Acceleration, and Force:

    System

    Mass

    Acceleration

    Force

    SI

    kilogram (kg)

    meter/second² (m/s²)

    newton (N)

    CGS

    gram (g)

    centimeter/second² (cm/s²)

    dyne (dyn)

    BE

    slug (sl)

    foot/second² (ft/s²)

    pound (lb)

  • Free-Body Diagram:

    • A free-body diagram represents an object and the forces acting on it.

  • Net Force Calculation Example:

    • For forces: 275 N + 395 N - 560 N: extNetForce=275N+395N−560N=+110Next{Net Force} = 275N + 395N - 560N = +110N

      • Direction: Directed along the +x axis.

  • Example of Applying Newton's Second Law to a Mass:

    • Example with a mass of the car =1850kg= 1850 kg,

      • Calculate acceleration where net force =110N= 110N:
        extAccelerationa=racFnetm=rac110N1850kg<br>ightarrowaextcalculationwouldyieldanaccuratenumericalvalue.ext{Acceleration} a = rac{F_{net}}{m} = rac{110N}{1850kg} <br>ightarrow a ext{ calculation would yield an accurate numerical value.}

4.4 The Vector Nature of Newton’s Second Law

  • Vector Components in Forces:

    • ∑Fx=mimesax∑F_{x} = m imes a_{x}

    • ∑Fy=mimesay∑F_{y} = m imes a_{y}

    • Analysis involves breaking down forces acting on an object into their x and y components.

  • Free-Body Diagram Example:

    • A raft example showing forces:

    • Forces acting: $P$, $A$, represented with components in several directions, showing equilibrium conditions.

  • Calculation Examples for Raft Net Force:

    • X Components: 17 N,

    • Y Components: calculated using trigonometric functions based on angles as shown in diagrams.

4.5 Newton’s Third Law of Motion

  • Statement:

    • Whenever one body exerts a force on a second body, the second body exerts an oppositely directed force of equal magnitude on the first body.

  • Example Considerations with Forces:

    • If a force is applied to a spacecraft and astronaut scenario detailing two differing masses, equal and opposite force computation resulting in forces of 36 N leads to calculated accelerations as follows:

      • Spaceship mass: 11,000 kg and acceleration computation.

      • Astronaut mass: 92 kg and acceleration based on the same force.

4.6 Types of Forces: An Overview

  • General Types of Forces in Nature:

    • Fundamental Forces:

      1. Gravitational force

      2. Strong Nuclear Force

      3. Electroweak Force

    • Nonfundamental Forces Examples:

      • Friction, tension in a rope, normal or support forces.

4.7 The Gravitational Force

  • Newton’s Law of Universal Gravitation:

    • Every particle in the universe exerts an attractive force on every other particle.

    • Each force is directed along the line joining the particles.

  • Force Calculation Between Two Particles:

    • The gravitational force magnitude formula:
      F=Gracm1m2r2F = G rac{m_1 m_2}{r^2}

    • G is the gravitational constant: G=6.673imes10−11Nimesm2/kg2G = 6.673 imes 10^{-11} N imes m^2/kg^2

  • Example Calculation of Gravitational Force:

    • Two particles with masses m1=25kgm_1 = 25 kg, m2=12kgm_2 = 12 kg separated by a distance of 1.2m1.2 m:
      F=Gracm1m2r2,extApplyingtheformulayieldsthegravitationalforceinNewtons.F = G rac{m_1 m_2}{r^2}, ext{ Applying the formula yields the gravitational force in Newtons.}

  • Definition of Weight:

    • Weight is defined as the gravitational force exerted on an object by the Earth, always acting downwards towards its center.

    • SI Unit of Weight:

      • Newton (N).

  • Relationship Between Mass and Weight:

    • Equation example:
      W=mimesgW = m imes g

    • Here, gg represents the acceleration due to gravity (approximately 9.81m/s29.81 m/s² on Earth).

  • Earth Surface Weight Calculation:

    • W=racGimesracmEimesmr2gW = rac{G imes rac{m_E imes m}{r^2}}{g}

4.8 The Normal Force

  • Normal Force Definition:

    • The normal force is a component of force exerted by a surface on an object in contact, specifically the component perpendicular to the surface.

  • Normal Force Example Calculation:

    • If a person stands on a scale, the reading reflects the normal force exerted upon them as their apparent weight.

  • Force Component Equation:

    • extstyle{oldsymbol{∑F_y} = F_N - mg = ma}

    • Describing the balance of forces in vertical dynamics, yielding insights into apparent weight versus true weight.

4.9 Static and Kinetic Frictional Forces

  • Definition:

    • When in contact with surfaces, a frictional force acts parallel to the surface which opposes relative motion.

    • Static Friction: Occurs when surfaces do not slide across each other.

  • Maximum Static Friction Formula:

    • Fsext(max)=extµsimesFNF_s ext{ (max)} = ext{µ_s} imes F_N

    • 0 < µ_s < 1 known as the coefficient of static friction.

  • Kinetic Friction:

    • Occurs when the objects are sliding past one another.

    • fk=µkFNf_k = µ_k F_N, with 0 < µ_k < 1 .

  • Approximate Values Table of Coefficients of Friction:

    • Ex: Glass on glass has a coefficient of static friction 0.94 and kinetic friction 0.4.

  • Examples Illustrating Frictional Forces in Action:

    • Scenarios where static friction controls, including sled motions against surfaces demonstrating kinetic friction properties.

4.10 The Tension Force

  • Definition:

    • Tension is a force that is transmitted through ropes or cables often modeled as massless for simplification.

  • Massless Rope Rule:

    • Tension is unchanged on both ends of a massless rope if passed around a frictionless pulley.

4.11 Equilibrium Application of Newton’s Laws of Motion

  • Definition of Equilibrium:

    • An object is said to be in equilibrium when there is zero acceleration:

      • ∑F=0∑F = 0 and ∑τ=0∑τ = 0

  • Reasoning Strategy for Equilibrium:

    1. Select objects to apply equilibrium equations.

    2. Draw a free-body diagram indicating forces acting on the object.

    3. Choose x, y axes, and resolve all forces into components.

    4. Apply equilibrium equations to solve unknowns.

  • Equilibrium Example Scenarios:

    • Numerous examples demonstrating forces, angles, and equilibrium analysis through free-body diagrams.

4.12 Nonequilibrium Application of Newton’s Laws of Motion

  • Non-Equilibrium Dynamics:

    • When an object is accelerating, it is not in equilibrium.

    • Such that, ∑Fx=mimesax∑F_{x} = m imes a_{x} and ∑Fy=mimesay∑F_{y} = m imes a_{y} govern the analysis.