Energy and Work

TYPES OF ENERGY

Spring Potential Energy

• Basic idea: Potential energy due to spring force.
• Hooke’s Law force: Fx = -k(x - x{eq})
  • Where:
    • F_x is the spring force.
    • k is the spring constant.
    • x is the displacement from equilibrium.
    • x_{eq} is the equilibrium position.
• Spring Potential Energy: PE{spring} = \frac{1}{2}k(x - x{eq})^2
  • Spring constant k (N/m or N/cm).
  • Stretch x - x_{eq} from equilibrium in the x-direction.

Clicker Question

A spring-loaded projectile launcher launches a ball straight up. Comparing the moment right before launch to the highest point:

• Which type of energy did not change?
  • Answer: C) Spring potential energy of the spring

Example Problem: Compressed Spring

• Initially, a 200 N/cm spring is uncompressed with a 10 cm length.
• A 400 N force is applied, compressing it to a shorter length.
  • How long is the spring while compressed?
    • 8 cm
  • How much spring potential energy is stored in the spring?
    • 4 J

Class Problem: Scale Energy

• A 9 kg watermelon is placed on a spring-supported platform with a 1500 N/m spring. After a while, it settles down and stops.
  • How far does the spring compress?
    • 5.88 cm
  • What is the change in spring potential energy?
    • 2.59 J
  • What is the change in the gravitational potential energy of the watermelon?
    • -5.19 J

Internal Energy

• Objects are composed of atoms and molecules.
  • “Stationary” objects have microscopic motion.
  • “Isolated” objects have internal spring-like bonds.
• Hidden energy is called “internal energy”.
  • Hidden KE: Molecular motion.
  • Hidden PE: Spring-like bonds.
• Definition: Energies that can’t be measured by macroscopic positions and speeds.
• Internal energy changes indicated by:
  • Phase changes: solid < liquid < gas.
  • Temperature changes: hotter = more energy.
  • Chemical bond changes: engines, muscles, chemical reactions & plastic deformation.

Clicker Question

• In which scenario is there no change in internal energy?
  • Answer: D) A person picks up a box and puts it on a shelf

Mechanical Energy

• Definition:

  • Energy that can be calculated from macroscopic positions and speeds.

• Mechanical energies:

  • Kinetic energy (KE).
    • KE{mech} = \frac{1}{2}m{ball}v_{ball}^2
    • With the speed of the whole object, not individual atoms.
  • Spring potential energy.
    • With actual spring, rather than “spring like” chemical bonds.
  • Gravitational potential energy.

• Friction converts mechanical energy into internal energy.

  • Energy can’t be entirely converted back to mechanical energy.

Clicker Question

• A ball is thrown across the room. Is its kinetic energy mechanical or internal energy?
  • Answer: A) Mechanical

TRANSFERS OF ENERGY

Work in General

• Basic idea: A transfer of energy by pushing over a distance.
  • “Work on object” = energy transfer into object.
• For a constant force:
  • W = F \Delta r cos \theta
  • Where:
    • W is Work
    • F is Magnitude of applied force
    • \Delta r is Magnitude of displacement of object being pushed
    • \theta is the Angle between F and \Delta r

Clicker Question

• A constant 200 N force pushes a cart toward the left while the cart moves directly upward 2 m. What is the work done on the cart by the force?
  • Answer: E) 0 J

Equivalent Formulas for Work

• In terms of a component of force
  • Positive \F_\parallel for \theta < 90^\circ
  • Negative \F_\parallel for \theta > 90^\circ
• In terms of a component of displacement
  • Positive \Delta x_\parallel for \theta < 90^\circ
  • Negative \Delta x_\parallel for \theta > 90^\circ
• W = F_\parallel \Delta r
  • Component of force in direction of displacement
• W = F \Delta x_\parallel
  • Component of displacement in the direction of force
• In terms of all components:
  • Works for any choice of x & y directions
  • W = Fx\Delta x + Fy\Delta y + F_z\Delta z

Example Problem: Calculating Work

A constant 200 N force pushes a cart toward the left while the cart moves directly to the right 350 cm to the right then 50 cm to the left.

• What is the work done on the cart by the force?
  • Answer: -600 J
    ** Calculate with each formula for work.

Math Aside: Dot Product

• Basic idea:
  • One of two ways to multiply vectors
  • Result is a scalar
• Notation
  • “Dot” like A⋅B
  • No implicit multiplication (never write AB)
  • Not the same as regular multiplication A⋅B ≠ AB
• Equivalent formulas
  • A \cdot B = ABcos\theta
  • A \cdot B = A_\parallel B
  • A \cdot B = AB_\parallel
  • A \cdot B = AxBx + AyBy + AzBz
  • Can’t do trick of replacing arrow with component!
  • A = 5B does imply Ax = 5Bx
  • A \cdot B = 5 does not imply AxBx = 5

Clicker Question

• Which of the following is an equivalent expression for F \cdot v?
  • Answer: E) None of the above

Work with Dot Product

• Basic idea:
  • A shorthand for any of the equivalent formulas
• For a constant force:
  • W = F \cdot \Delta r
    • Applied force
    • Displacement of object being pushed

Clicker Question

• A person holds a 5 kg book above their head for 1 s without moving. How much work did they do on the book?
  • Answer: A) 0 J

Clicker Question

• A person holds a 5 kg book above their head for 1 s in an elevator lowering at 1 m/s. How much work did they do on the book?
  • Answer: E) -50 J

Clicker Question

• An athlete pushes a 7 kg shot put over 1 m with 500 N of force. How much work did they do on the shot put?
  • Answer: B) 500 J

Work From Varying Forces

• Basic idea
  • Work is the area under the curve of a force vs distance graph
  • Area above axis = + work
  • Area below axis = - work
  • F_\parallel Component of force parallel to velocity (tangent to path)
  • x Distance along path

Work for Specific Forces

Direction of force based on velocity

• Constraint force
  • Some tension and normal forces
• Kinetic friction
• Associated with potential energy
  • Gravitational force
  • Spring forces

Constraint Forces

• Basic idea:
  • Forces that simply keep an object on a path
• Definition:
  • Forces that are always perpendicular to velocity
• Examples:
  • Normal forces keeping an object on a stationary track
• Constraint forces do no work!
  • W_{constraint} = 0

Clicker Question

A 3,000 kg roller coaster slides down a 10 m tall, frictionless track.

• How much work did the track do on the roller coaster?
  • Answer: A) 0 J

Work from Kinetic Friction

• Requires
  • Magnitude of normal force remains constant
  • Wk = -\mukF_NL
    • Work done by a kinetic friction force
    • Coefficient of kinetic friction
    • Normal force
    • Length of path traveled along surface

Class Problem: Braking Work

An 1,800 kg car going 70 mph (31.3 m/s) hits the brakes and skids to a stop. The coefficient of kinetic friction between the tire and pavement is 0.8.

• Calculate the change in kinetic energy of the car as it stops.
  • -881 kJ
• Calculate the normal force by the road on the car.
  • 17.6 kN
• If all the kinetic energy is transferred to the ground via work by the kinetic friction, find the distance it takes to come to a complete stop.
  • 62.4 m = 205 ft

Example Problem: Braking on a Hill

A 2000 kg car going 70 m/s brakes to a stop while driving down a hill with a 10° angle from horizontal. It takes the car 400 m to come to a stop.

• A) Find the change in the car’s kinetic energy.
  • -4.90 MJ
• B) Find the change in the car’s gravitational potential energy
  • -1.36 MJ
• C) Find the work done by the road on the car, assuming that the kinetic and gravitational potential energy was all transferred to the road via kinetic friction.
  • -6.26 MJ
• D) Find the coefficient of kinetic friction between the car tires & road.
  • 0.811

Class Problem: Sliding Down

An 8 kg box slides down a ramp over 80 cm, as shown, with a 0.2 coefficient of kinetic friction between the box and ramp. Find:

• The magnitude of the normal force of the ramp on the box.
  • 71.1 N
• The work done by the ramp on the box.
  • -11.4 J
• The change in gravitational potential energy of the box.
  • -26.5 J
• The change in the kinetic energy of the box, assuming all gravitational potential energy not lost to the box via work goes into the kinetic energy of the box.
  • 15.1 J
• The speed of the box at the bottom, assuming the box starts at rest?
  • 1.95 m/s

Technical Details: About Work

• Work “done on” vs work “done by”
  • Work done on A = how much energy flows into object A
  • Work done by A = how much energy flows out of object A
  • Negative work = opposite direction
    • W{on A} = 5 J means W{by A} = -5 J
    • W{on A} = -5 J means W{by A} = 5 J
• Detailed Definition of Work
  • Work needs more subscripts
    • Force has “by” & “on” objects and type
    • Displacement of the “on” object
  • W{by A on B} = F{by A on B}\Delta r_B cos \theta
  • W{by B on A} = F{by B on A}\Delta r_A cos \theta
• Work “On” vs Work “By”
  • So far, our description of work implies
  • W{on A} = -W{by A}
• But is this true?
  • If the objects do not get closer or farther apart then YES
  • Otherwise NO
• Why?
  • Same by Newton’s 3rd Law.
  • Opposite Why

Work and Potential Energy

• What if W{by B on A} ≠ -W{by A on B}?
  • Happens when
    • A & B change separation while pushing or pulling on each other
  • Interpretation
    • Positive value = energy entering “on” object
    • Negative value = energy leaving “on” object
• Where does the energy go (or come from)?
  • Potential energy
• Money Analogy for Work and Potential Energy
  • Potential energy is a joint account
    • Held jointly by A & B
    • Often phrased as energy of one object (esp. 𝑃𝐸𝑔)
    • One account for each conservative force
    • Gravitational or spring force
  • Work by that force is a withdrawal
    • W_{by A on B} is a withdrawal by B
    • W_{by B on A} is a withdrawal by A
    • Positive = withdrawal, negative = deposit
• Work and Potential Energy Formulas
  • Work produces changes in potential energy
    • Sign flip (Negative work = increase in PE)
    • W{by B on A} + W{by A on B} = -\Delta PE
• Work and Stationary Objects
  • Often one object is stationary (\Delta r_B = 0)
    • Earth stationary for PE_g
    • One side of the spring stationary for PE_{spring}
  • Stationary objects have no work done on them
  • Can have work done by a stationary object
  • Stationary objects easier work formula
    • W_{by B on A} = -\Delta PE
• Why stationary object does not move?

Work for Specific Forces: Part 2

• Work by Gravitational Forces
  • Basic idea
    • Find work using potential energy formulas
  • Gravitational force work
    • Wg = -\Delta PEg
• Class Problem: Work on the Stairs
  • A 60 kg person moves from the ground to the top of a 443 m tall building. Find the work done on the person by the Earth via gravity.
    • Answer: -260 kJ
• Work by Spring Forces
  • Basic idea
    • Find work using potential energy formulas
  • Spring force work
    • Energy stored within spring
    • W{spring} = -\Delta PE{spring}
• Class Problem: Pinball Machine
  • An 80.6 g pinball is shot along an inclined surface from a compressed spring with a 3 N/cm spring constant and 15 cm equilibrium length
    • Find the work done by the spring until launched.
      • 0.585 J
    • Find the gravitational work on the pinball until launched.
      • -8.1 x 10^{-3} J
    • Assuming the work transfers the energy into the pinball’s kinetic energy, find the launch speed.
      • 3.78 m/s
    • Find the gravitational work on the pinball from launch until top.
      • -0.135 J
    • Assuming the work transfers the energy into the pinball’s kinetic energy, find the speed at the top.
      • 3.31 m/s
• Work by Friction

Basic idea

• Friction “loses” energy into internal energy
• Difficult to notice changes in internal energy
• Technical details
  • This is for “nonconservative” forces
  • Appears to not conserve energy
  • Friction is our example of a nonconservative force
  • Other examples:
    • Normal forces causing plastic deformation (dents)
    • Tension causing plastic deformation (broken springs)

Transfers of Energy: Summary

• General formulas
  • W = F\Delta r cos \theta
  • W = F_\parallel \Delta r
  • W = F\Delta x_\parallel
  • W = Fx\Delta x + Fy\Delta y
  • W = F \cdot \Delta r
• Constraint forces
  • W_{constraint} = 0
• Kinetic friction
  • Wk = -\mukF_NL
• Spring forces
  • W{spr} = -\Delta PE{spring}
• Gravitational
  • Wg = -\Delta PEg

Other Methods of Transferring Energy: Hidden Work

• Mechanical waves (sound, earthquakes, etc.)
  • Forces within the substance
• Heat flow
  • Spontaneously flow of energy due to temperature differences
  • Atoms pushing on each other in collisions
• Electrical transmission
  • Electric forces pushing on electrons

Other Methods of Transferring Energy: Stuff With Energy Moves

• Matter transfer
  • Example: convection
• Electromagnetic radiation
  • Examples: light, microwaves, radio waves, etc.

Power

• Basic idea:
  • The rate at which energy flows
• Definition
  • SI unit:
    • Watt (W = J/s)
  • P = \frac{dE}{dt}
  • Sign indicates direction
    • + → energy flows into object’s kinetic energy
    • - → energy flows out of object’s kinetic energy
• Finding Power

Power via work

• Rate of energy flow into object via force

  • P = Fvcos\theta = F\parallel v = Fv\parallel = Fxvx + Fyvy + Fzvz
    • F Magnitude of force on object
    • v Speed of object
    • \theta Angle between F & v
  • P = F \cdot v
    • F: Force on object
    • v: Velocity of object
      Seeing Where Energy Goes

• Energy flows into object’s KE through force

  F & v < 90° apart

• Energy flows out of object’s KE through force

  F & v > 90° apart

• No energy flow through force

  F & v perpendicular, not moving, or no force

Clicker Question

How does the force on the object affect its kinetic energy?

• Answer: B) It is currently decreasing its kinetic energy

Visualizing Flows of Energy

• Indications of Changing Energy
  • Kinetic energy → speed
    • Faster → more KE
  • Gravitational potential energy → height
    • Higher → more PEg
  • Spring potential energy → spring length
    • More stretched or compressed → more PEspring
  • Internal energy → temperature change, phase change, plastic deformation (dents), chemical reactions, muscle or motor work
• Indications of Energy Flow
  • Moving in same direction as applied force → gain KE via force
  • Moving in opposite direction as applied force → lose KE via force
  • Stationary or move perpendicular to applied force → no energy transfer via force
  • Where is the energy coming from or going to?
    • 𝑃𝐸 with “by” object if Fg or F{spring}
    • To internal energy for friction, mostly
    • From internal energy for engine, motors, or muscle work
    • Otherwise, “by” object’s kinetic energy
      Changes of Energy
• PE_{spring} (length changes)
• Mass’s KE (speed changes)
  Energy Flow
• Wall stationary → no energy flow
• Mass & spring moving → transfer energy between PE_{spring} and Mass’s KE

Clicker Question

In what direction does energy flow between the ball and bat while in contact?

• Answer: B) Energy flows from the bat to the ball

Clicker Question

In what direction does energy flow between the pole and the ground while they are in contact?

• Answer: C) No energy flows between the pole and ground

Clicker Question

In what direction does energy flow between the pole and the athlete while he is rising?

• Answer: A) Energy flows from the athlete to the pole

Conservation of Energy

Systems and Environments

• System
  • Basic idea
    • A list of object you want to keep track of
  • More precisely, a list of energy “accounts”
  • System vs environment
    • A division of the universe into “inside” (system) or “outside” (environment)
    • Everything not in the system is in the “environment”
• Closed (or Isolated) Systems

Definition

• A system with no net work into or out of it.
• Visualization
  • No work lines cross system boundary
    Situations with Closed (or Isolated) Systems
    No external forces on system
• No work possible
• Everything in system
  A system where all external forces are constraint forces or gravitational forces
• Hockey pucks on ice
• Frictionless rollercoaster
• Bouncing ball
• Open Systems

Definition

• A system with net work into or out of it.
• Visualization
  • Work lines cross system boundary
    Situations with Open Systems
    Net force on system
• A ball hit by a bat
  No net force, but net work
• Crushing a soda can
  Choosing Your System
  Good objects to include in system
• Start with objects you want to know about
  • Speed, height, spring compression
• Add objects interacting with objects in system that you do know about
  • Speed, height, spring compression
• Add enough objects to close the system, if possible

Clicker Question

A roller coaster runs along a track with negligible friction. If the system is just the roller coaster including its 𝑃𝐸𝑔, is it closed?

• Answer: A) Yes

Clicker Question

A roller coaster runs along a track with negligible friction. If the system is just the roller coaster not including its 𝑃𝐸𝑔, is it closed?

• Answer: B) No