Energy Conservation and Momentum

Total Energy of a System

Conservation of Energy

Total Energy of a System, E_{tot}

  • Definition:
    • Sum of all types of energy for all objects in a system.
    • E_{tot} can be positive or negative.
    • Kinetic Energy (KE) must be positive.
    • Potential Energy (PE) can be positive or negative.

Visualizing Motion on PE vs x Graphs

  • Basic idea:
    • Visualize the range of motion and relative speed.
  • Scenario:
    • One object moving in 1D.
  • What to draw:
    • Total potential energy as a curve.
    • Total energy as a horizontal line.
  • Key elements on the graph include:
    • PE(x) representing potential energy as a function of position.
    • x_{eq} representing the equilibrium position.
    • E_{tot} representing total energy.
    • The range of motion defined by the region where E_{tot}
      geq PE(x).

Visualizing Motion on PE vs x Graphs

  • Seeing relative speed:
    • Kinetic energy is E_{tot} - PE.
    • The farther the PE curve is below E_{tot}, the faster the object moves.
  • Range of motion:
    • Kinetic energy can't be negative.
    • The object stays where PE < E_{tot}.
  • New term: "Bound system":
    • Finite range of motion of objects.

Energy Conservation

Equations

Change in Total Energy of a System, \Delta E_{tot}

  • Definition:
    • Sum of changes in energy in a system during some time interval.
  • Find by noticing all the energies that are changing in a system.
  • Example 1: From highest point to just before impact
    • \Delta E{tot} = \Delta PE{g,ball} + \Delta KE_{ball}

Change in Total Energy of a System, \Delta E_{tot}

  • Definition:
    • Sum of changes in energy in a system during some time interval.
  • Find by noticing all the energies that are changing in a system.
  • Example 2: From squished to highest point
    • \Delta E{tot} = \Delta PE{g,ball} + \Delta PE_{spring,ball}

Conservation of Energy in Systems

  • General equation:
    • \Delta E{tot} = \sum W{ext}
      • \Delta E_{tot}: Change in total energy of system
      • \sum W_{ext}: Total work from external forces
  • How to use:
    • Choose a system.
    • Choose a time interval.
    • Replace \Delta E_{tot} with the sum of all energy changes in the system during the time interval.
    • Replace \sum W_{ext} with the sum of work by external objects on objects in the system during the time interval.

Example Energy Conservation Equation: Closed System

  • Time interval:
    • Initial = highest point
    • Final = just before impact
  • System: Earth + Ball
  • Environment: Floor
  • Equation:
    • \Delta PEg + \Delta KE{ball} = 0
    • \Delta E{tot} = \sum W{ext}
    • mg\Delta y + {1 \over 2} m (vf^2 - vi^2) = 0

Example Energy Conservation Equation: Open System

  • Time interval:
    • Initial = before impact
    • Final = just after impact
  • System: Ball
  • Environment: Bat
  • Equation:
    • \Delta KE{ball} = W{by Bat on Ball}
    • {1 \over 2} m{ball} (v{f,ball}^2 - v{i,ball}^2) = W{by Bat on Ball}
    • \Delta E{tot} = \sum W{ext}

Work or Potential Energy?

  • W{g, by E on ball} = \Delta PE{g,ball}

  • Equivalent expressions:

    • \Delta PE{g,ball} + \Delta KE{ball} = 0
    • mg\Delta y + {1 \over 2} m (vf^2 - vi^2) = 0
    • {1 \over 2} m (vf^2 - vi^2) = -mg\Delta y

  • Don't double count!

  • Including PE_g in the system means you don't include the work done by gravity as an external force.

  • Not including PE_g in the system means you must include the work done by gravity as an external force.

Problem Solving Steps

Conservation of Energy

Problem Solving Steps With Energy Conservation

  1. Identify a system
  2. Identify a time interval
    • Initial: highest point, final: stopped at bottom
  3. Identify energy changes in a system
    • Gravitational PE and spring PE
  4. Set up energy conservation equation
    • \Delta E{total} = \sum W{ext}
    • \Delta PEg + \Delta PE{spring} = 0

Problem Solving Steps With Energy Conservation

  1. Plug in formulas for each type of energy
  2. Solve for variable of interest
    • mg (yf - yi) + {1 \over 2} k (xf - x{eq})^2 - (xi - x{eq})^2 = 0
    • xf = x{eq} - \sqrt{{2mg \over k} y_i}

Issues with Internal Energy

CONSERVATION OF ENERGY

Issues with Internal Energy

  • Basic idea:
    • Internal energy is hidden, so \Delta E_{int} is hard to calculate from the current state.
  • Solution:
    • Calculate with work from non-conservative forces (friction).
    • \Delta E{int} = -W{friction}

Reworking Energy Conservation Equation

  • Start with conservation of energy equation:
    • \Delta E{tot} = \sum W{ext}
  • Split total energy into mechanical and internal:
    • \Delta E{tot,mech} + \Delta E{tot,internal} = \sum W_{ext}
  • Replace internal energy with work:
    • \Delta E{tot,mech} + (-\sum W{friction}) = \sum W_{ext}
  • Rewritten to have work on one side:
    • \Delta E{tot,mech} = \sum W{friction} + \sum W_{ext}

Conservation of Mechanical Energy in Systems

  • General equation:
    • \Delta E{tot,mech} = \sum W{friction} + \sum W_{ext}
      • \Delta E_{tot,mech}: Change in total mechanical energy of system
      • \sum W_{friction}: Total work from friction (technically, nonconservative forces)
      • \sum W_{ext}: Total work from external forces
  • How to use:
    • Choose a system & time interval.
    • Replace \Delta E_{tot,mech} with mechanical energies changing in the system during the time interval.
    • Replace \sum W_{ext} with work by external objects during the time interval.
    • Replace \sum W_{friction} with work by frictional (nonconservative) forces even by objects in the system.
    • Don’t include the same work twice for both W{friction} and W{ext}.

Example Energy Conservation Equation: Mechanical Energy

  • Time interval:
    • Initial = just before skid
    • Final = right after stopping
  • \Delta KE{car} = W{k,by ground on car}
    • {1 \over 2} m vf^2 - {1 \over 2} m vi^2 = -\muk FN L
    • \Delta E{tot,mech} = \sum W{friction} + \sum W_{ext}

Momentum & Collisions

GENERAL PHYSICS I LECTURE

Slides Overview

  • Introduction to Momentum
    • What is momentum
    • Momentum vs kinetic energy
  • Impulse
    • Transferring momentum
  • Conservation of Momentum
    • Conservation equations
  • Collisions
    • Elastic & inelastic collisions
  • Center of Mass
    • Use in other formulas
    • Using to find motion

Introduction to Momentum

Momentum, \vec{p}

  • Momentum is a property of an object
    • Roughly, how hard it is to stop an object
  • Notes
    • Also called “linear momentum”
    • No special name for units (kg \cdot m/s)
    • Vector not just a number
  • \vec{p} = m\vec{v}
    • \vec{p}: momentum
    • m: mass
    • \vec{v}: velocity

Important Features of Momentum

  • Momentum is transferred via forces.
    • The amount transferred is called impulse.
  • Momentum only relates to motion
    • No "potential momentum"
  • Momentum cannot be hidden like energy
    • No "internal momentum"
  • Momentum cannot be created or destroyed

Momentum vs Kinetic Energy

FeatureMomentumKinetic Energy
IncludesDirection and speedSpeed only
DescribesHow hard it is to stopHow much damage it can do
Formula\vec{p} = m\vec{v}KE = {1 \over 2} mv^2
KE relationshipKE = {p^2 \over 2m}

Impulse

Impulse, \vec{J}

  • Basic idea:
    • Pushing over a time.
  • For a constant force:
    • \vec{J} = \vec{F} \Delta t
      • \vec{J}: Impulse
      • \vec{F}: Force
      • \Delta t: Duration of time interval
  • Impulse is a transfer of momentum
    • \vec{J}_{by A on B} is the momentum transferred from A to B
    • Analogous to work

Impulse from a Varying Force

  • Impulse = area under curve on F_x vs t
    • Jx for Fx vs t and Jy for Fy vs t
    • Positive for area above the axis, negative below
    • Area under F_x vs t curve
  • Or use average force in formula
    • \vec{J} = \vec{F}_{avg} \Delta t
      • \vec{J}: Impulse
      • \vec{F}_{avg}: Force
      • \Delta t: Duration of time interval
  • Hard to have \vec{J} = 0 if \vec{F} \neq 0
    • Balanced + & - area, or
    • Very small approximated as zero

Conservation of Momentum Equations

Conservation of Momentum in Systems

  • General equation:
    • \Delta \vec{p}{tot} = \sum \vec{J}{ext}
      • \Delta \vec{p}_{tot}: Change in total momentum of system
      • \sum \vec{J}_{ext}: Total impulse from external forces
  • How to use:
    • Choose a system
    • Choose a time interval
    • Replace \Delta \vec{p}_{tot} with the sum of all momentum changes in the system during the time interval
    • Replace \sum \vec{J}_{ext} with the sum of impulses by external objects on objects in the system during the time interval

Systems in Energy vs Momentum Conservation

FeatureMOMENTUMENERGY
SystemList of objectsList of energy accounts
Closed systemNo net external impulse on system Effectively no net force on systemNo net external work on system

Collisions

What is a Collision?

  • Basic idea
    • A brief interaction causing hard forces
    • Motion determined by interaction
  • Rephrasing in physics language
    • An interaction between system of objects where the external impulses are far less than the internal impulses
    • \sum J{by:system:objects:on:A} >> \sum J{by:external:objects:on:A}

Momentum Conservation in Collisions

  • Basic idea
    • Momentum is always conserved in collisions
  • Reason
    • Collision definition is good approx. of \vec{p} conservation condition
    • \sum J{by:system:objects:on:A} >> \sum J{by:external:objects:on:A}
    • \sum J{by:system:objects:on:A} + \sum J{by:external:objects:on:A} = \Delta p_1
    • Negligible \approx 0

Impulse in Collisions

  • Utility
    • Can find impulse even when forces & duration are unknown
  • Common confusion
    • “Impulse” often used to mean momentum transferred in negligible time

Elastic & Inelastic Collisions

  • Basic idea
    • Kinetic energy is conserved in “bouncy” collisions
  • Elastic collisions
    • Kinetic energy conserved
    • Perfectly bouncy, so objects return to their original shape
    • \Delta KE_{total} = 0
  • Inelastic collisions
    • Kinetic energy is not conserved
    • Dents, sound, or deformations, so objects are changed after the collision
    • \Delta KE_{total} \neq 0

What is Conserved?

Collision TypeKinetic EnergyMomentum
Elastic\Delta KE_{total} = 0\Delta p = 0
Inelastic\Delta KE_{total} \neq 0\Delta p_{total} = 0
*Technically, mechanical energy is conserved in elastic collisions, but in practice, that means kinetic energy.

Conservation Equations in 1D Elastic Collisions

  • Momentum conservation
    • \Delta p_{total} = 0
    • m1(v{1f} - v{1i}) + m2(v{2f} - v{2i}) + … = 0
  • Mechanical energy conservation
    • \Delta KE_{total} = 0
    • {1 \over 2} m1 (v{1f}^2 - v{1i}^2) + {1 \over 2} m2 (v{2f}^2 - v{2i}^2) + … = 0

Conservation Equations in 1D Inelastic Collisions

  • Momentum conservation
    • \Delta p_{total} = 0
    • m1(v{1f} - v{1i}) + m2(v{2f} - v{2i}) + … = 0
  • No kinetic energy conservation
    • \Delta KE_{total} \neq 0
  • If they stick together, they have the same v_f
    • v{1f} = v{2f} = …

Center of Mass

Center of Mass, \vec{r}_{CM}

  • Basic idea
    • “The center” or “balance point” of an object or set of objects
  • More precisely
    • Divide object(s) into equal mass chunks & take the average position
    • \vec{r}{CM} = (x{CM}, y_{CM})

Center of Mass Equation

  • Requirements
    • Point masses, or extended objects using their center of mass position
    • No equation for finding the center of mass of an extended object
    • Near the middle usually but requires calculus for exact answer
  • Center of mass equation
    • \vec{r}{CM} = {\sumi mi \vec{r}i \over m_{total}}
    • x{CM} = {m1 x1 + m2 x2 + … \over m1 + m_2 + …}
    • y{CM} = {m1 y1 + m2 y2 + … \over m1 + m_2 + …}

Center of Mass & Motion

  • So far, every part of an object must have the same velocity
    • Point particle or rigid & non-rotating
  • For any object, use m_{total} and center of mass positions in equations
    • Except kinetic energy
    • Using center of mass speed gives only “translational” kinetic energy
    • Also, “rotational” kinetic energy and kinetic energy in internal energy
  • \sum \vec{F} = m \vec{a} \rightarrow \sum \vec{F} = m{total} \vec{a}{CM}
  • \vec{p}{total} = m{total} \vec{v}_{CM}
  • KE{trans} = {1 \over 2} m{total} v_{CM}^2
  • PEg = mgy \rightarrow PEg = m{total} gy{CM}

Center of Mass & Momentum

  • The total momentum of a system uses
    • Center of mass velocity
    • Total mass
  • The center of mass velocity is constant for any closed system!
  • Can relate center of mass motion to motion of individual objects:
    • \vec{p}{total} = m{total} \vec{v}_{CM}
    • m{total} \Delta x{CM} = m1 \Delta x1 + m2 \Delta x2 + …
    • m{total} \Delta y{CM} = m1 \Delta y1 + m2 \Delta y2 + …

Center of Mass & Motion

  • Center of mass motion is simple!
    • Collisions
      • Center of mass moves at constant velocity

Center of Mass & Motion

  • Center of mass motion is simple!
    • Collisions
      • Center of mass moves at constant velocity
    • Orbits
      • Center of mass moves at constant velocity

Center of Mass & Motion

  • Center of mass motion is simple!
    • Collisions
      • Center of mass moves at constant velocity
    • Orbits
      • Center of mass moves at constant velocity
    • Projectile Motion
      • Center of mass moves at in parabola

Static Equilibrium & Torque

GENERAL PHYSICS I

Slides Overview

  • Describing 2D rotational motion
    • Angular variables
    • Angular kinematic equations
    • Relating angular and linear quantities
  • Torque
    • Introducing torque
    • Center of gravity
  • Static equilibrium
    • Static conditions
    • Problem-solving with static equilibrium
    • Types of equilibrium

Describing Rotational Motion in 2D

Rigid Body Motion

  • To find the location of every part you only need:
    • Center of mass location
    • Orientation
  • Tools to find center of mass location
    • \sum \vec{F} = m{total} \vec{a}{CM}
    • \vec{p}{total} = m{total} \vec{v}_{CM}
    • KE{trans} = {1 \over 2} m{total} v_{CM}^2
    • PEg = m{total} gy_{CM}
  • How to find orientation?

Describing Orientation in 2D with Angles

  • What is an angle?
    • The ratio of the arc length to radius
  • Angles are unitless
    • “Radian” = reminder of definition
    • “Degree” = \pi/180
  • Angles have a sign
    • Counterclockwise = +, clockwise = -
    • Same convention for all angular quantities
  • \theta = s/r

Angular Quantities

Angular QuantityVariableUnits
Angle\thetaNone
Angular Displacement\Delta \theta = \thetaf - \thetaiNone
Angular Velocity\omega = {d\theta \over dt}1/s
Angular Acceleration\alpha = {d\omega \over dt}1/s^2

Angular vs. Linear

Angular QuantityLinear Quantity
AnglePosition
Angular DisplacementDisplacement
Angular VelocityVelocity
Angular AccelerationAcceleration

Kinematic Equations

Angular Equation if \alpha is constantLinear Equation If a is constant
\omegaf = \omegai + \alpha tvf = vi + at
\omegaf^2 = \omegai^2 + 2 \alpha \Delta \thetav_f^2 = v + 2a \Delta x
\omega{avg} = {\omegai + \omega_f \over 2}v{avg} = {vi + v_f \over 2}
\thetaf = {1 \over 2} \alpha t^2 + \omegai t + \theta_ixf = {1 \over 2} a t^2 + vi t + x_i

Changing Frames

  • Generally, switching origin location changes all angular quantities
    • Angle, \theta
    • Angular velocity, \omega
    • Angular acceleration, \alpha
    • Quantities not yet learned: Torque, moment of inertia, angular momentum
  • Need to specify origin
    • Phrased as angle “about” a point
  • Which origin should you choose?

Pivot Points as Origin

  • Basic idea
    • The location that doesn’t move in a rotation
  • Created by hinge or equivalent
    • Applies necessary force to keep pivot point stationary
  • Choose as origin when possible
    • Same angular velocity for each part of a rigid object
    • A much simpler description

Center of Mass as Origin

  • Use center of mass when there is rotation but no pivot point
    • Object thrown in the air
    • Hinge or equivalent spot moves
  • Why?
    • Thrown objects pivot around center of mass
    • Avoids confusing effects
    • Fictitious forces causing fictitious torques