chapter 5 – work and energy. 5.2 mechanical energy

Chapter 5 – WORK and ENERGY

5.2 MECHANICAL ENERGY

ENERGY = ability to do WORK

ENERGY = ability to do WORK

the energy of an object that is due to the

object’s motion

KINETICPOTENTIALthe energy

associated with an object

because of its position, shape,

or condition

KINETIC ENERGY

A cart:mass = mforce =

constantSo a =

constant

Δx

Wnet = F ∙ dWnet = F ∙ ΔxWnet = m ∙ a

∙ Δx

KINETIC ENERGY

Δx

Wnet = m ∙ a ∙ Δx

Wnet = ½ mvf2 – ½

mvi2

KINETIC ENERGY depends on SPEED and MASS

KINETIC ENERGY = KE = Ek =

Wnet = ½ mvf2 – ½

mvi2

KINETIC ENERGY depends on SPEED and MASS

KINETIC ENERGY = KE = Ek =

Wnet = ½ mvf2 – ½ mvi

2 = ΔKE = ΔEk

WORK – KINETIC ENERGY THEOREM

Net work = change in kinetic energy

Wnet = ½ mvf2 – ½

mvi2 Wnet = KEf – KEi

Wnet = ΔKE

KINETIC ENERGY = KE = Ek =

UNITS = Joules

KINETIC ENERGY depends on SPEED and MASS

WORKSHEET EXAMPLEA 6.0 kg cat runs after a mouse at

10 m/s. What is the cat’s kinetic

energy?

WORKSHEET EXAMPLEA 6.0 kg cat runs after a mouse at

10m/s. What is the cat’s kinetic

energy?KE = 300 J

Wnet = 4.1975 x 104 J = 4.2 x 104 J

Wnet = ½ mvf2 – ½ mvi

2 = ΔKE

Wnet = 7.2 x 10-2 J

EXAMPLEOn a frozen pond, a person kicks a 10.0 kg sled, giving it an initial speed of 2.2 m/s.

How far does the sled move if the coefficient of kinetic friction between

the sled and the ice is 0.10?

EXAMPLEOn a frozen pond, a person kicks a 10.0 kg sled, giving it an initial speed of 2.2 m/s.

How far does the sled move if the coefficient of kinetic friction between

the sled and the ice is 0.10?GIVEN

?UNKNOWN

?FBD?

mass = 10.0 kgvi = 2.2

m/svf = 0.0

m/sμk = 0.10

d = Δx = ??

EXAMPLEmass = 10.0 kgvi = 2.2

m/svf = 0.0

m/sμk = 0.10

d = Δx = ??Wnet = F ∙ d

Net Wnet is due to Ff, so…

Wnet = Ff ∙ d

-½ mvi2 = μk ∙ m ∙ g ∙ dd = 2.5 m

WORKSHEET EXAMPLEHow much net work is needed to

accelerate a 1000.0 kg car from 20.0 m/s to 30.0 m/s

WORKSHEET EXAMPLEHow much net work is needed to

accelerate a 1000.0 kg car from 20.0 m/s to 30.0 m/s

HOMEWORKWorksheetProblems: Practice B

Practice C

19

POTENTIAL ENERGY

gravitational

elastic

POTENTIAL ENERGYthe energy associated with an object

because of its position, shape, or condition

It is stored energy

POTENTIAL ENERGYDepends on the properties of an

object and its interactions with its environment

It is stored energy

GRAVITATIONAL POTENTIAL ENERGYIs the energy of an object due to the

objects position relative to a gravitational source

i.e. energy stored in an object due to its position relative to the Earth’s

gravitational field

GRAVITATIONAL POTENTIAL ENERGYi.e. energy stored in an object due to

its position relative to the Earth’s gravitational field

Potential energy

Kinetic energy

Where did the KE come from?

GRAVITATIONAL POTENTIAL ENERGYi.e. energy stored in an object due to

its position relative to the Earth’s gravitational field

Potential energy

Kinetic energy

GRAVITATIONALPOTENTIAL

ENERGY

GRAVITATIONAL POTENTIAL ENERGYi.e. energy stored in an object due to

its position relative to the Earth’s gravitational field

Ep = PEg = Ug = m ∙ g ∙ h UNITS =

JoulesREMEMBER:a. free – fall acceleration = constant

(near the E’s surface) for PEg to be valid

b. It is relative (because measuring the height is arbitrary)

TRUE or FALSE?Suppose I throw a volleyball from a second floor roof (A), and it lands on the first floor

of an adjacent building (B)…

If the height is measured from the ground (C),

PEg is zero.

PEg = m∙g∙h

If the height is measured from the 1st floor (B),

PEg is zero.

TRUE

FALSE

Can You Have a Negative PE?

If YES, give an example

If NOT, explain why.

Can You Have a Negative PE?

You could bring a textbook from a table height to a zero – level (ground) –

performing a negative work

If yes, give an example

If not, explain why.

Can you have an object with a positive PE relative to one point and negative PE to another point at the same time?

If YES, give an example

If NOT, explain why.

A textbook which is 0.5 m below a table (negative PE relative to the

table)

If yes, give an example

If not, explain why.

A textbook which is 0.5 m above the ground (positive PE relative to

the table)

Can you have an object with a positive PE relative to one point and negative PE to another point at the same time?

m = 25 kg

h = 6 m

Potential energy doubles

ELASTIC POTENTIAL ENERGYIs the energy of available for use when a deformed elastic object

(spring, bungee cord…) returns to its original position

PEelastic = ½ kx2

x

PEelastic = ½ spring constant ∙ (distance compressed/stretched)2

k = SPRING CONSTANTMeasures how easy is to compress or

stretch a spring

SMALL k LARGE kVery flexible

Easy to stretch

Very stiffDifficult to

stretch

WHAT IS THE UNIT OF k?

WORKSHEET EXAMPLEA 70.0 kg stuntman is attached to a bungee cord with an

unstretched length of 15.0 m. He jumps off a bridge spanning a river from a height of 50.0 m. When he finally

stops, the cord has a stretched length of 44.0 m. Treat the stuntman as a point mass, and disregard the weight

of the bungee cord and air resistance. Assuming the spring constant of the bungee cord is 71.8 N/m,

WHAT IS THE TOTAL POTENTIAL ENERGY RELATIVE TO THE WATER WHEN THE MAN

STOPS FALLING?

EXAMPLEA 70.0 kg stuntman is attached to a bungee cord with an

unstretched length of 15.0 m. He jumps off a bridge spanning a river from a height of 50.0 m. When he finally

stops, the cord has a stretched length of 44.0 m. Treat the stuntman as a point mass, and disregard the weight

of the bungee cord and air resistance. Assuming the spring constant of the bungee cord is 71.8 N/m,

WHAT IS THE TOTAL POTENTIAL ENERGY RELATIVE TO THE WATER WHEN THE MAN

STOPS FALLING?PEtot = PEg + PEelastic

EXAMPLEA 70.0 kg stuntman is attached to a bungee cord with an

unstretched length of 15.0 m. He jumps off a bridge spanning a river from a height of 50.0 m. When he finally

stops, the cord has a stretched length of 44.0 m. Treat the stuntman as a point mass, and disregard the weight of the bungee cord. Assuming the spring constant of the

bungee cord is 71.8 N/m, What is the total potential energy relative to the water when

the man stops falling?

GIVEN?

UNKNOWN?

FBD?

Choose the water level to be 0 PEg

EXAMPLEmass = 70.0

kg

k = 71.8 N/m

h = 50.0 – 44.0 = 6.0 m

x = 44.0 – 15.0 = 29.0 m

PE = 0 J (at river level)

PEtot = PEg + PEelastic = ?

h Choose the water level to be 0 PEg

x

EXAMPLEmass = 70.0

kg

k = 71.8 N/m

h = 50.0 – 44.0 = 6.0 m

x = 44.0 – 15.0 = 29.0 m

PE = 0 J (at river level)

PEtot = PEg + PEelastic = ? PEg = m ∙ g ∙ hPEelastic = ½ kx2

PEtot = 3.43 x 104 J

WORKSHEET EXAMPLEWhen a 2.00 kg mass is attached to a vertical spring, the spring is stretched 10.0 cm such that the mass is 50.0

cm above the table.a.What is the gravitational

potential energy associated with this mass relative to the table?

b.What is the spring’s elastic potential energy if the spring

constant is 400.0 N/m?

c.What is the total potential energy of this system?

WORKSHEET EXAMPLEWhen a 2.00 kg mass is attached to a vertical spring, the

spring is stretched 10.0 cm such that the mass is 50.0 cm above the table.

a. What is the gravitational potential energy associated with this mass relative to the table?

b. What is the spring’s elastic potential energy if the spring constant is 400.0 N/m?

c. What is the total potential energy of this system?

HOMEWORKWorksheet:Problems: Practice D

43

MECHANICAL ENERGY

Chemical, nuclear,

electrical…

ME = KE + PE

MECHANICAL ENERGY

PENDULUM

MECHANICAL ENERGY

Describe each picture in terms of potential and kinetic

energy

MECHANICAL ENERGY

PEg present (at max)

No KE present

PEg presentKE present

No PEg present

KE present (at max)

PEg present (at max)

No KE present

PEg present KE present

No PEg present

KE present (at max)

MECHANICAL ENERGY

MECHANICAL ENERGY

ME = KE + ∑PE

CONSERVATION OF ENERGY

Who can describe what is

happening in the next

demonstration?

ME is often conserved E remains

constant

KEi + ∑ PEi = KEf + ∑ PEf

CONSERVATION OF ENERGY

KEi + ∑ PEi = KEf + ∑ PEf

CONSERVATION OF ENERGY

EXAMPLEStarting from rest, a child zooms down

a frictionless slide from an initial height of 3.00 m. WHAT IS HER SPEED

AT THE BOTTOM OF THE SLIDE? Assume she has a mass of 25.0 kg.

EXAMPLEStarting from rest, a child zooms down

a frictionless slide from an initial height of 3.00 m. WHAT IS HER SPEED

AT THE BOTTOM OF THE SLIDE? Assume she has a mass of 25.0 kg. GIVEN

?UNKNOWN

?FBD?

vi = 0.0

m/sh = hi = 3.00 mhf = 0.00

m

vf = ? m/s

EXAMPLEStarting from rest, a child zooms down

a frictionless slide from an initial height of 3.00 m. WHAT IS HER SPEED

AT THE BOTTOM OF THE SLIDE? Assume she has a mass of 25.0 kg.

FBD?PEg,i = m ∙ g ∙ hi

PEg,f = 0 J

KEi = 0 J

KEf = ½ mvf

2

KEi + ∑PEi = KEf + ∑PEf

vf = 7.67 m/s

WORKSHEET EXAMPLEA small 10.0 g ball is held to a

slingshot that is stretched 6.0 cm. The spring constant is 2.0 × 102 N/m.

WORKSHEET EXAMPLEA small 10.0 g ball is held to a

slingshot that is stretched 6.0 cm. The spring constant is 2.0 × 102 N/m.a. What is the elastic potential energy of the slingshot before

it is released?b. What is the kinetic energy

of the ball just after the slingshot is released?

c. What is the ball’s speed at that instant?

d. How high does the ball rise if it is shot directly upward?

WORKSHEET EXAMPLEA small 10.0 g ball is

held to a slingshot that is stretched 6.0

cm. The spring constant is 2.0 × 102

N/m.a. What is the elastic

potential energy of the slingshot before it is

released?b. What is the kinetic energy of the ball just after the slingshot is

released?c. What is the ball’s

speed at that instant?d. How high does the ball rise if it is shot

directly upward?

HOMEWORKProblems: Practice E

60

QUIZDate: Tuesday, Dec 17

What?: Chapter 5.2 Mechanical Energy

61

ANSWERS TO

WORKSHEET EXAMPLES

WORKSHEET EXAMPLEA 6.0 kg cat runs after a mouse at

10m/s. What is the cat’s kinetic

energy?KE = 300 J

WORKSHEET EXAMPLEHow much net work is needed to

accelerate a 1000.0 kg car from 20.0 m/s to 30.0 m/s

Can You Have a Negative PE?

You could bring a textbook from a table height to a zero – level (ground) –

performing a negative work

If yes, give an example

If not, explain why.

Can you have an object with a positive PE relative to one point and negative PE to another point at the same time?

A textbook which is 1.0 m below a table (negative PE relative to the

table)

If yes, give an example

If not, explain why.

A textbook which is 1.0 m above the ground (positive PE relative to

the table)

WHAT IS THE UNIT OF k?

N/m

WORKSHEET EXAMPLEWhen a 2.00 kg mass is attached to a vertical spring, the

spring is stretched 10.0 cm such that the mass is 50.0 cm above the table.

a. What is the gravitational potential energy associated with this mass relative to the table?

b. What is the spring’s elastic potential energy if the spring constant is 400.0 N/m?

c. What is the total potential energy of this system?