Chapter 6: Work, Energy and Power Thursday February 12th

Reading: up to page 97 in the text book (Ch. 6)

• Discuss Mini Exam II • Review: Work and Kinetic Energy • Conservative and non-conservative forces • Work and Potential Energy • Conservation of Energy • Calculus method for determining work • Power • As usual – iclicker, examples and demonstrations

Mini Exam III next Thursday • Will cover LONCAPA #7-10 (Newton’s laws and energy cons.)

Work - Definition

• There are only two relevant variables in one dimension: the force, Fx, and the displacement, Δx.

Work W is the energy transferred to or from an object by means of a force acting on the object. Energy transferred to the object is positive work, and energy transferred from the object is negative work.

W = FxΔxDefinition: [Units: N.m or Joule (J)]

Fx is the component of the force in the direction of the object’s motion, and Δx is its displacement.

Kinetic Energy - Definition

Fx Frictionless surface

Δx

K = 12 mv 2

12 mv f

2 − 12 mvi

2 = 12 m× 2axΔx

ΔK = K f − Ki = maxΔx = FxΔx =W

v f2 = v i

2+2axΔx

Work-Kinetic Energy Theorem

ΔK = K f − Ki =Wnet

change in the kinetic net work done on

energy of a particle the particle

⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

K f = Ki +Wnet

kinetic energy after kinetic energy the net

the net work is done before the net work work done

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Fx = F cosφW = Fd cosφW =

!F ⋅!d

More on Work To calculate the work done on an object by a force during a displacement, we use only the force component along the object's displacement. The force component perpendicular to the displacement does zero work

• Caution: for all the equations we have derived so far, the force must be constant, and the object must be rigid.

• I will discuss variable forces later.

Work – More Examples

k M

K=½mv2

Frictionless surface

M

K=0 k

Δx

Fsp.

Frictionless surface

Work – More Examples

Wspr. = −FΔx = ΔK

k M

K=½mv2

Δx

Fsp.

Frictionless surface

Kinetic energy is completely recovered: a ‘conservative’ force

Work – More Examples

Wspr. = +FΔx = ΔK

F

Mg

h

Wlift = FhWg = −Mgh

These two examples are similar – they both involve conservative forces Examples: electrostatic (spring/elastic) forces, gravitational forces.

Work – More Examples

Wn.c. = Fh = +MghWcons. =Wg = −MghWnet =Wn.c. +Wcons. = 0

If F = Mg,Wnet = 0Then ΔK = 0

F

Mg

h

It turns out that one can define a ‘Potential Energy’, U, for ALL conservative forces as follows:

Work & Potential Energy

ΔU =U f −Ui

= −Wcons.

ΔUg = Mgh=Wn.c.

F

Mg

ΔU =U f −Ui

= −Wcons.

ΔUg = Mgh=Wn.c.

The Potential Energy change, ΔU, does not care how the height change was achieved.

h

It turns out that one can define a ‘Potential Energy’, U, for ALL conservative forces as follows:

Work & Potential Energy

Conservation of Energy

ΔK = K f − Ki =Wnet

=Wcons. +Wn.c.

Work-Kinetic Energy theorem

• We can now replace any work due to conservative forces by potential energy terms, i.e.,

ΔK = −ΔU +Wn.c.

ΔEmech = ΔK + ΔU =Wn.c.

Or

• Here, Emech is the total mechanical energy of a system, equal to the sum of the kinetic and potential energy of the system.

• If work is performed on the system by an external, non-conservative force, then Emech increases.

Conservation of Energy Special case: a completely isolated system, solely under the influence of conservative forces:

ΔEmech = ΔK + ΔU = 0

⇒ K f − Ki( ) + U f −Ui( ) = 0

Or K f +U f = Ki +Ui

If there is an outside influence from a non-conservative force, then:

K f +U f = Ki +Ui( ) +Wn.c.

Power • Power is defined as the "rate at which work is done." • If an amount of work W is done in a time interval Δt by a force, the average power due to the force during the time interval is defined as

Pavg =

WΔt

• Instantaneous power is defined as P = dW

dt• The SI unit for power is the Watt (W).

1 watt = 1 W = 1 J/s = 0.738 ft · lb/s 1 horsepower = 1 hp = 550 ft · lb/s = 746 W

1 kilowatt-hour = 1 kW · h = (103 W)(3600 s) = 3.60 MJ

General (calculus) method for calculating Work

ΔU = −Wc

= − Fc1Δx + Fc2Δx + ....FcjΔx + ...( )= − FcjΔx

j∑

ΔU = −Wc = − Fc (x)dx

xi

x f∫

Take the limit Δx → 0

Elastic/Spring Potential Energy

ΔU = − (−kx)dxxi

x f∫ = k x dxxi

x f∫= 1

2 k x2⎡⎣ ⎤⎦xi

x f = 12 kx f

2 − 12 kxi

2

ΔU = −Wc = − Fc (x)dx

xi

x f∫F

xi xf

Fc = -kx Δx

Fc = kx

U = − (−kx)dx0

x

∫ = k x dx0

x

∫= 1

2 k x2⎡⎣ ⎤⎦0

x= 1

2 kx2

0

x

-kxf