AE Version 2.0 11/09/18.

Always, sometimes, never?

AE Version 2.0 11/09/18.

Work and

energy

Avril Steele

Toby Rome

Simon Clay

AE Version 2.0 11/09/18.

Work & energy

AE Version 2.0 11/09/18.

Topics using energy Work, energy & power

• Definitions, work-energy principle,

conservation of energy

Elastic springs & strings

• Hooke’s law & elastic potential energy

Circular motion

• Vertical circular motion

Momentum & restitution

• Loss of kinetic energy in collisions

AE Version 2.0 11/09/18.

Specifications referencesAQA Edexcel OCR (A) OCR(B)

MEI

Work, energy

& power AS level FM1 AS AS level Mech a

Elastic springs

& stringsAS level FM1 A level Mech b

Circular

motionA level FM2 AS level Mech b

Momentum &

restitutionAS level FM1 AS AS level Mech a

AE Version 2.0 11/09/18.

Definitions

Kinetic Energy KE =

Gravitational Potential

EnergyGPE =

Elastic Potential Energy EPE =

Work Done WD =

𝐹 = force

𝑚 = mass

𝑣 = speed

𝑠 = distance moved

ℎ = height

𝜆 = modulus of elasticity

𝑥 = extension

𝑙 = natural length

𝑔 = acceleration due to gravity

AE Version 2.0 11/09/18.

Definitions

Kinetic Energy 𝐾𝐸 =1

2𝑚𝑣2

Gravitational Potential

Energy𝐺𝑃𝐸 = 𝑚𝑔ℎ

Elastic Potential Energy 𝐸𝑃𝐸 =𝜆𝑥2

2𝑙

Work Done 𝑊𝐷 = 𝐹𝑠

𝐹 = force

𝑚 = mass

𝑣 = speed

𝑠 = distance moved

ℎ = height

𝜆 = modulus of elasticity

𝑥 = extension

𝑙 = natural length

𝑔 = acceleration due to gravity

AE Version 2.0 11/09/18.

The work-energy principleHow do we get from Newton’s second law to the

work-energy principle… …and to conservation of

energy?

𝐹 = 𝑚 × 𝑎

Could be

variableLet’s keep

this constant

This is a rate

of change

AE Version 2.0 11/09/18.

The work-energy principle

𝐹 = 𝑚𝑎 = 𝑚d𝑣

d𝑡= 𝑚

d𝑥

d𝑡

d𝑣

d𝑥= 𝑚𝑣

d𝑣

d𝑥

𝐹 d𝑥 = 𝑚𝑣 d𝑣

Work done =1

2𝑚𝑣2

𝑣1

𝑣2

Work done =1

2𝑚𝑣2

2 −1

2𝑚𝑣1

2

𝑥 is the

displacement

Work done = Change in KE

AE Version 2.0 11/09/18.

Work doneBy a constant force

𝐹 moving a particle

a distance 𝑠 in the

direction of the

force

0

𝑠

𝐹 d𝑥 = 𝐹 𝑥 0𝑠 = 𝐹𝑠

By the weight of an

object as it moves

from height ℎ1 to

height ℎ2 (relative

to some datum)

ℎ1

ℎ2

−𝑚𝑔 d𝑥 = −𝑚𝑔 𝑥 ℎ1ℎ2 = −(𝑚𝑔ℎ2 −𝑚𝑔ℎ1)

By the tension

in a spring

stretching from

an extension 𝑥1to an extension 𝑥2

𝑥1

𝑥2

−𝜆𝑥

𝑙d𝑥 = −

𝜆

𝑙

1

2𝑥2

𝑥1

𝑥2

= −𝜆𝑥22

2𝑙−𝜆𝑥12

2𝑙

𝐹𝑠

𝑚𝑔ℎ1

ℎ2

𝑙 + 𝑥1

𝑙 + 𝑥2

AE Version 2.0 11/09/18.

The work-energy principle

𝐾𝐸 =1

2𝑚𝑣2 𝐺𝑃𝐸 = 𝑚𝑔ℎ 𝐸𝑃𝐸 =

𝜆𝑥2

2𝑙

Work done = change in KE + change in GPE + change in EPE

by ‘other’ forces

by all forces

Work done = Change in KE

𝐹𝑠 − (𝑚𝑔ℎ2 −𝑚𝑔ℎ1)−𝜆𝑥22

2𝑙−𝜆𝑥12

2𝑙=1

2𝑚𝑣2

2 −1

2𝑚𝑣1

2

AE Version 2.0 11/09/18.

The work-energy principle

𝐹𝑠 =1

2𝑚𝑣2

2 −1

2𝑚𝑣1

2 + (𝑚𝑔ℎ2 −𝑚𝑔ℎ1)+𝜆𝑥22

2𝑙−𝜆𝑥12

2𝑙

𝐾𝐸 =1

2𝑚𝑣2 𝐺𝑃𝐸 = 𝑚𝑔ℎ 𝐸𝑃𝐸 =

𝜆𝑥2

2𝑙

Work done = change in KE + change in GPE + change in EPE

by ‘other’ forces

by all forces

Work done = Change in KE

AE Version 2.0 11/09/18.

Conservation of energy

If there are no ‘other’ forces doing work then the total

change in energy is zero. Energy is conserved.

Work done = change in KE + change in GPE + change in EPE

AE Version 2.0 11/09/18.

AE Version 2.0 11/09/18.

Dropping a ball

What

assumptions are

we making?

AE Version 2.0 11/09/18.

Bungee jumperWhere should the release point be so that the tape

just touches the floor?

For red elastic

take 𝜆 = 6.5 N?

AE Version 2.0 11/09/18.

Calculations

Conservation of energy:

𝜆𝑥2

2𝑙= 𝑚𝑔(𝑙 + 𝑥)

𝜆𝑥2

2𝑙−𝑚𝑔𝑥 −𝑚𝑔𝑙 = 0

𝑙 + 𝑥

𝐾𝐸 = 0, 𝐺𝑃𝐸 = 𝑚𝑔 𝑙 + 𝑥 , 𝐸𝑃𝐸 = 0

𝐾𝐸 = 0, 𝐺𝑃𝐸 = 0, 𝐸𝑃𝐸 =𝜆𝑥2

2𝑙

release point

lowest point

The total distance fallen is 𝑙 + 𝑥, and

so the mass should be released from

rest at this height above the surface,

so that it just contacts the surface as it

comes to rest.

Don’t forgot to account for the size of

the object though!

AE Version 2.0 11/09/18.

Circular loops What’s the relationship between OA and OB for

the bob to complete a full circle about B?

AE Version 2.0 11/09/18.

Energy in circular motionCondition for particle on string to make full circle is

that 𝑢 > 5𝑔𝑟 at A.

At A: 𝐺𝑃𝐸 = 0, 𝐾𝐸 =1

2𝑚𝑢2

At B: 𝐺𝑃𝐸 = 2𝑚𝑔𝑟, 𝐾𝐸 =1

2𝑚𝑣2

O

A 𝑢 ms-1

𝑟 m

B

See handout for more

information on circular

motion.

AE Version 2.0 11/09/18.

Calculations

𝐾𝐸 = 0, 𝐺𝑃𝐸 = 𝑚𝑔𝑙

𝐾𝐸 =1

2𝑚(5𝑔𝑟), 𝐺𝑃𝐸 = 0

Conservation of energy:

5

2𝑚𝑔𝑟 = 𝑚𝑔𝑙

⇒ 𝑟 =2

5𝑙

The radius of the large arc is 𝑙.The radius of the small circle is 𝑟.

𝑂𝐵 = 𝑙 − 𝑟 ⇒ 𝑂𝐵 =3

5𝑂𝐴

𝑟

AE Version 2.0 11/09/18.

Where will it land?Is this statement justified?

Work done by friction = change in KE + change in GPE

AE Version 2.0 11/09/18.

Mechanics in actionElasticity

• Dangerous sports club (sheet 22)

Circular motion

• Looping-the-loop 1 & 2 (sheets 27 & 37)

• Wind up (sheet 38)

Collisions

• Bouncing balls 1 & 2 (sheets 49 & 50)

stem.org.uk/resources/elibrary/resource/26065/mechanics-action

2019 FM Conference

V1.1

Version 1.0

1 of 1

Work and energy

Specifications

AQA Edexcel OCR (A) OCR(B) MEI

Work, energy & power AS level FM1 AS AS level Mech a

Elastic springs & strings AS level FM1 A level Mech b

Circular motion A level FM2 AS level Mech b

Momentum & restitution AS level FM1 AS AS level Mech a

Definitions

2