Newton's Laws

Physics

  • Physics is the basic science concerned with the nature of basic things such as:
    • Motion
    • Forces
    • Energy
    • Matter
    • Heat
    • Sound
    • Light
    • Structure of atoms

Dynamics

  • Dynamics is a field in classical mechanics that studies the relationships among forces, matter, and motion.
  • Force is defined as a push or a pull.
  • It is an interaction between bodies or an object with its surroundings.
  • Force is a vector quantity with SI unit called Newtons.

Fundamental Forces of Nature

  • Strong Force:
    • Binds atomic nuclei together.
  • Electromagnetic Force:
    • Facilitates electric or magnetic interactions.
    • Binds molecules together.
  • Weak Force:
    • Enables radioactive decay.
  • Gravitational Force:
    • Attracts bodies with mass together.
  • Arranged from strongest to weakest.

Forces in Everyday Life

  • Walking to class: Electromagnetic force
  • Dropping a phone: Gravitational force
  • Accelerating in a motorized vehicle: Electromagnetic force

Force as a Vector

  • A force can be broken down into its components.

Common Types of Forces

  • Contact Forces: Interactions occurring upon contact.
    • Normal Force ($\vec{n}$):
      • Opposing force exerted by the surface on an object.
      • Always perpendicular to the plane of the surface.
    • Frictional Force ($\vec{f}$):
      • Opposes the relative motion with respect to the surface.
      • Exerted on an object by a surface and acts parallel to the surface.
    • Tensional Force ($\vec{T}$):
      • A pulling force exerted by and acting along a stretched string/rope or cable.
  • Long-range Forces: Interactions occurring without contact.
    • Weight ($\vec{w}$):
      • The gravitational attraction between the Earth and an object.
      • Always directed downward.
    • Other examples: Electric and Magnetic Forces
  • Applied Force:
    • Forces not defined earlier and have an indefinite source.
    • Can be contact or long-range.

Superposition of Forces

  • Any number of forces applied at a point on an object have the same effect as a single force equal to the vector sum of the forces, also known as the resultant force or the net force.

Vector Addition

  • Two vectors can be added to form a resultant vector.
  • Addition can be done using the tail-to-head method.
  • $\vec{C} = \vec{A} + \vec{B}$

Scalar Multiplication

  • A scalar can be multiplied to a vector to change its magnitude.
  • Allows us to negate a vector and do vector subtraction.

Addition of Vectors: Components

  • Given the vectors:
    • $\vec{A} = Ax\hat{i} + Ay\hat{j} + A_z\hat{k}$
    • $\vec{B} = Bx\hat{i} + By\hat{j} + B_z\hat{k}$
  • Performing a vector summation is simply:
    • $\vec{A} + \vec{B} = (Ax + Bx)\hat{i} + (Ay + By)\hat{j} + (Az + Bz)\hat{k}$
      *Note: Components can be negative.

Subtraction of Vectors: Components

  • Likewise, for vector subtraction:
    • $\vec{A} - \vec{B} = (Ax - Bx)\hat{i} + (Ay - By)\hat{j} + (Az - Bz)\hat{k}$
      *Note: Components can be negative.

Example 2

  • Three forces are acting on an object. The magnitudes of the forces are F<em>1=250 NF<em>1 = 250 \text{ N}, F</em>2=50.0 NF</em>2 = 50.0 \text{ N}, and F3=120 NF_3 = 120 \text{ N}.
  • The x and y components of the net force:
    • $\vec{F}{\text{net}} = \vec{F}1 + \vec{F}2 + \vec{F}3$
    • $\vec{F}_1 = -150 \text{ N } \hat{i} + 200 \text{ N } \hat{j}$
    • $\vec{F}_2 = 50.0 \text{ N } \hat{i}$
    • $\vec{F}_3 = -120 \text{ N } \hat{j}$
    • $\vec{F}_{\text{net}} = (-150 \text{ N } + 50.0 \text{ N }) \hat{i} + (200 \text{ N } + -120 \text{ N }) \hat{j}$
    • $\vec{F}_{\text{net}} = -100 \text{ N } \hat{i} + 80 \text{ N } \hat{j}$
  • The magnitude and direction of the net force:
    • Fnet=128 NF_{net} = 128 \text{ N}
    • θ=141∘, wrt the +x-axis\theta = 141^\circ \text{, wrt the +x-axis}

Mass vs. Weight

  • Mass:
    • Intrinsic property of an object.
    • Related to the compactness of an object's molecules and molecular mass.
    • Constant anywhere in the universe.
  • Weight:
    • Gravitational force on a body depends on the object’s mass and the acceleration due to gravity.

Example 3

  • An object is falling at a rate of 3.78 m/s23.78 \text{ m/s}^2 near the surface of Mars.
  • By how much will the object's mass and magnitude of weight increase on the surface of Earth?
    • gMars=3.78 m/s2g_{\text{Mars}} = 3.78 \text{ m/s}^2
    • gEarth=9.8 m/s2g_{\text{Earth}} = 9.8 \text{ m/s}^2
    • Object mass is the same in Mars and in Earth.
    • Object mass not needed for calculation.
    • w<em>Earthw</em>Mars=mg<em>Earthmg</em>Mars\frac{w<em>{\text{Earth}}}{w</em>{\text{Mars}}} = \frac{mg<em>{\text{Earth}}}{mg</em>{\text{Mars}}}
    • w<em>Earthw</em>Mars=g<em>Earthg</em>Mars=2.59\frac{w<em>{\text{Earth}}}{w</em>{\text{Mars}}} = \frac{g<em>{\text{Earth}}}{g</em>{\text{Mars}}} = 2.59
    • w<em>Earth=2.59w</em>Marsw<em>{\text{Earth}} = 2.59 w</em>{\text{Mars}}
    • Object mass the same anywhere

Newton's First Law

  • "Every object continues in a state of rest or of uniform speed in a straight line unless acted on by a nonzero net force."
  • "An object acted on by no net external force has a constant velocity (which may be zero) and zero acceleration."
  • "A body at rest remains at rest or, if in motion, remains in motion at constant velocity unless acted on by a net external force."
  • F⃗<em>net=F⃗</em>1+F⃗<em>2+⋯+F⃗</em>N=0\vec{F}<em>{\text{net}} = \vec{F}</em>1 + \vec{F}<em>2 + \dots + \vec{F}</em>N = 0
  • Breaks intuition that a force is required to sustain motion
  • Sometimes referred to as the principle of inertia
  • Inertia - the tendency of an object to maintain its state of motion (including being at rest)

Example 4

  • You are driving a Maserati GranTurismo S on a straight testing track at a constant speed of 250 km/h250 \text{ km/h}. You pass a 1972 Volkswagen Beetle doing a constant 75.0 km/h75.0 \text{ km/h}.
  • On which car is the net force greater?
    • Both have zero acceleration.
    • Hence, both have zero net force.

Equilibrium

  • An object is in equilibrium if it is either at rest or moves at constant velocity (no change in magnitude and direction).
  • Equilibrium happens when all forces cancel out; the net force is zero.

Mass and Force

  • The proportionality between the net force and acceleration implies a constant of proportionality.
  • The constant of proportionality turns out to be the object's mass.

Newton’s Second Law

  • "If a net external force acts on a body, the body accelerates. The direction of acceleration is the same as the direction of the net force. The mass of the body times the acceleration of the body equals the net force vector."
  • In symbols we have, F⃗net=ma⃗\vec{F}_{\text{net}} = m\vec{a}
  • Note that the net force is the sum of all external forces acting object
  • Acceleration is not proportional to A FORCE; it is proportional to the NET FORCE! Account for all forces acting on the object ∑<em>i=1NF⃗</em>i=ma⃗\sum<em>{i=1}^{N} \vec{F}</em>i = m\vec{a}

Using Newton's Second Law

  • We calculate the net force doing multiple one-dimensional calculations
  • Each component will have an equation of the form: F<em>net,x=ma</em>xF<em>{net,x} = ma</em>x with similar forms for the y and z components
  • Newton's Second Law refers to external forces acting on an object
    • The equation below is only valid if the mass is constant

Example 5

  • A worker applies a constant horizontal force with magnitude 20 N20 \text{ N}, to a box with mass 40 kg40 \text{ kg} resting on a level floor with negligible friction. What is the acceleration of the box?
    • Fnet=maF_{net} = ma
    • a=Fnetma = \frac{F_{net}}{m}
    • F=20 N i^F = 20 \text{ N } \hat{i}
    • m=40 kgm = 40 \text{ kg}
    • ax=0.50 m/s2i^a_x = 0.50 \text{ m/s}^2 \hat{i}
    • ay=0a_y= 0
    • az=0a_z= 0

Example 6

  • A dockworker applies a constant horizontal force with magnitude 80 N80 \text{ N} to a block of ice on a horizontal level floor. The frictional force is negligible. The block starts from rest and moves 11 m11 \text{ m} in 5.0 s5.0 \text{ s}. What is the mass of the block of ice?
    • Δx=12at2\Delta x = \frac{1}{2}at^2
    • m=F<em>neta=F</em>nett22Δxm = \frac{F<em>{net}}{a} = \frac{F</em>{net} t^2}{2\Delta x}
    • F=80 N i^F= 80 \text{ N } \hat{i}
    • Δx=11 m\Delta x = 11 \text{ m}
    • t=5.0 st = 5.0 \text{ s}
    • m=91 kgm = 91 \text{ kg}

Example 7

  • A waitress shoves a ketchup bottle with mass of 0.450 kg0.450 \text{ kg} to her right along a smooth, level lunch counter. The bottle leaves her hand at 2.80 m/s2.80 \text{ m/s}, then slows down as it slides due to constant horizontal friction exerted on it by the counter top. It slides for 1.00 m1.00 \text{ m} before coming to rest. What are the direction and magnitude of the friction force acting on the ketchup bottle?
    • v<em>f2−v</em>i2=2aΔxv<em>f^2 - v</em>i^2= 2a\Delta x
    • F<em>net,x=ma=−mv</em>i22ΔxF<em>{net, x} = ma = -m\frac{v</em>i^2}{2\Delta x}
    • m=0.450 kgm = 0.450 \text{ kg}
    • Δx=1.00 m\Delta x = 1.00 \text{ m}
    • vi=2.80 m/sv_i = 2.80 \text{ m/s}
    • vf=0.00 m/sv_f = 0.00 \text{ m/s}
    • Fnet,x=−1.76 NF_{net, x} = -1.76 \text{ N}
    • The negative sign means left

Force in Uniform Circular Motion

  • Puck moves at constant speed around circle.
  • At all points, the acceleration a and the net force ΣF point in the same direction—always toward the center of the circle.
  • Fnet=mv2RF_{net} = \frac{mv^2}{R}

Newton's Third Law

  • If body A exerts a force on body B (an “action”), then body B exerts a force on body A (a “reaction”). These two forces have the same magnitude but are opposite in direction. These two forces act on different bodies.
  • The equal sign signifies that the magnitudes are equal
  • The negative sign indicates that they have opposite directions
  • Sometimes they are referred to as “action-reaction” pairs
  • F⃗<em>A on B=−F⃗</em>B on A\vec{F}<em>{A \text{ on } B} = -\vec{F}</em>{B \text{ on } A}

Newton's Third Law - Action and reaction pairs

  • Equal in magnitude
  • Opposite in directions
  • Acting on different bodies

Example 8

  • After your sports car breaks down, you start to push it to the nearest repair shop. While the car is moving, how does the force you exert on the car compare to the force of the car exerts on you? How do these forces compare when you are pushing the car at a constant speed?
    • All action-reaction pairs have equal magnitudes but opposite directions.

Example 9

  • An apple sits at rest on top of a table, in equilibrium. What forces act on the apple? What is the reaction force to each of the forces acting on the apple? What are the action-reaction pairs?

Example 10

  • A stonemason moves a marble block across a floor by pulling on a rope attached to the block. The block is not necessarily in equilibrium. How are the various forces related? What are the action-reaction pairs?

Class Exercise

  • (A) According to Newton's Third Law, forces come in pairs. While you are sitting on your chair, what is the force paired with your weight?
    • Gravitational force I exert on Earth
  • (B) Which of Newton's Laws of Motion would best explain
    • 1. …why a bowling bowl sliding along the bowling alley will have a constant velocity?
      • Law of Inertia
    • 2. …why a car slows down when you step on the brakes?
      • Law of Acceleration
    • 3. …why your head hurts when you accidentally hit it on the table?
      • Law of Action-Reaction
    • 4. …how thrust is generated by expelling propellants downward in order to make a space shuttle accelerate upward?
      • Law of Action-Reaction
    • 5. …why the ball’s direction changes when it bounces off a wall.
      • Law of Acceleration