Summary Lecture 7Summary Lecture 7

7.1-7.67.1-7.6 Work and Kinetic energyWork and Kinetic energy

8.28.2 Potential energyPotential energy

8.38.3 Conservative Forces and Conservative Forces and Potential energyPotential energy

8.58.5 Conservation of Mech. EnergyConservation of Mech. Energy

8.68.6 Potential-energy curvesPotential-energy curves

8.88.8 Conservation of EnergyConservation of Energy

Systems of ParticlesSystems of Particles

9.29.2 Centre of massCentre of mass

7.1-7.67.1-7.6 Work and Kinetic energyWork and Kinetic energy

8.28.2 Potential energyPotential energy

8.38.3 Conservative Forces and Conservative Forces and Potential energyPotential energy

8.58.5 Conservation of Mech. EnergyConservation of Mech. Energy

8.68.6 Potential-energy curvesPotential-energy curves

8.88.8 Conservation of EnergyConservation of Energy

Systems of ParticlesSystems of Particles

9.29.2 Centre of massCentre of mass

Problems: Chap. 8 5, 8, 22, 29, 36, 71, 51

Chap. 9 1, 6, 82

Problems: Chap. 8 5, 8, 22, 29, 36, 71, 51

Chap. 9 1, 6, 82

Thursdays 12 – 2 pm

PPP “Extension” lecture.

Room 211 podium level

Turn up any time

Thursdays 12 – 2 pm

PPP “Extension” lecture.

Room 211 podium level

Turn up any time

Outline Lecture 7Outline Lecture 7

Work and Kinetic energyWork and Kinetic energy

Work done by a net force results in kinetic energy

Some examples: gravity, spring, friction

Potential energyPotential energy

Work done by some (conservative) forces can be retrieved. This leads to the principle that energy is conserved

Conservation of EnergyConservation of Energy

Potential-energy curvesPotential-energy curves

The dependence of the conservative force on position is related to the position dependence of the PE

F(x) = -d(U)/dx

Kinetic Energy

Work-Kinetic Energy Theorem

Change in KE work done by all forces

K w

xixf

f

i

f

i

vv

vv vmdvvm ]/[. 221

= 1/2mvf2 – 1/2mvi

2

= Kf - Ki

= KK

Work done by net force

= change in KE

f

i

xx dxFw .

f

i

xx dxma .

f

i

f

i

xx

xx dv

dtdx

mdxdtdv

m ..

Work-Kinetic Energy TheoremF

x

Vec

tor

sum

of

all f

orce

s ac

ting

on

the

body

mg

F

h

Lift mass m with constant velocity

Work done by me (take down as +ve)

= F.(-h) = -mg(-h) = mghWork done by gravity

= mg.(-h) = -mgh ________

Total work by ALL forces (W) = 0

What happens if I let go?

=K

Gravitation and work

Work done by ALL forces = change in KE

W = K

Compressing a spring

Compress a spring by an amount x

Work done by me Fdx = kxdx = 1/2kx2

Work done by spring -kxdx =-1/2kx2

Total work done (W) =

0=K

What happens if I let go?

x

F -kx

Ff

dWork done by me = F.d

Work done by friction = -f.d = -F.d

Total work done = 0What happens if I let go? NOTHING!!

Gravity and spring forces are Conservative

Friction is NOT!!

Moving a block against friction at constant velocity

A force is conservative if the work it does on a particle that moves through a round trip is zero: otherwise the force is non-conservative

A force is conservative if the work done by it on a particle that moves between two points is the same for all paths connecting these points: otherwise the force is non-conservative.

Conservative Forces

A force is conservative if the work it does on a particle that moves through a round trip is zero; otherwise the force is non-conservative

Conservative Forces

work done by gravity for round trip:On way up: work done by gravity = -mgh

On way down: work done by gravity = mgh

Total work done = 0

Sometimes written as 0ds.F

h-g

Consider throwing a mass up a height h

Conservative Forces

-g

Each step height=h

= -mg(h1+h2+h3 +……)

= -mgh

Same as direct path (-mgh)

Work done by gravity

w = -mgh1+ -mgh2+-mgh3+…

h

A force is conservative if the work done by it on a particle that moves between two points is the same for all paths connecting these points: otherwise the force is non-conservative.

U = -w

Lift mass m with constant velocity

Work done by gravity

= mg.(-h) = -mgh

Potential EnergyThe change in potential energy is equal to minus the work done BY the conservative force ON the body.

Therefore change in PE is

U = -w

h

mg

Ugrav = +mgh

Work done by spring is w = -kx dx = - ½ kx2

Potential EnergyThe change in potential energy is equal to minus the work done BY the conservative force ON the body.

Therefore the change in PE is

U = - w

Compress a spring by an amount xF -kx

x

Uspring = + ½ kx2

Potential EnergyThe change in potential energy is equal to minus the work done BY the conservative force ON the body.

U = -wbut recall that

w = K so that

U = -K or

U + K = 0

Any increase in PE results from a decrease in KEdecrease n increase

U + K = 0In a system of conservative forces, any change in Potential energy is compensated for by an inverse change in Kinetic energy

U + K = EIn a system of conservative forces, the mechanical energy remains constant

Potential-energy diagrams

w= - U

The force is the negative gradient

of the PE curve

If we know how the PE varies with position, we can find the conservative force as a function of position

dx

dUF In the limit

x

UF

thus

= F. x

Energy

x

U= ½ kx2

kxF

dx

dUF

)kx(dx

ddx

dUFso

22

1

PE of a spring

F = -kx (spring force)

here U = ½ kx2

Energy

x

Potential energy

U= ½ kx2

U= ½ kA2

x=A

KE

PE

At any position x

PE + KE = E

U + K = E

K = E - U

= ½ kA2 – ½ kx2

= ½ k(A2 -x2)

x’

Total mech. energy

E= ½ kA2

Et

K

U

Fne

t=-d

U/d

t Fnet = mg – R

R = mg - Fnet

Roller Coaster

Et

K

U

Fne

t=-d

U/d

x Fnet = mg – R

R = mg - Fnet

mg

R

Conservation of Energy

We said: when conservative forces act on a body

U + K = 0 U + K = E (const)

This would mean that a pendulum would swing for ever.

In the real world this does not happen.

Conservation of Energy

When non-conservative forces are involved, energy can appear in forms other than PE and KE (e.g. heat from friction)

U + K + Uint = 0

Ki + Ui = Kf + Uf + Uint

Energy converted to other forms

Energy may be transformed from one kind to another in an isolated system, but it cannot be created or destroyed.

The total energy of the system always remains constant.

h

f mg

upward

Stone thrown into air, with air resistance. How high does it go?

Ei = Ef + Eloss

Ki + Ui = Kf + Uf + Eloss

½mvo2 + 0 = 0 + mgh + fh

½mvo2 = h(mg + f)

f)2(mgmv

h2

0

v0

h

f

mg

downward

Stone thrown into air, with air resistance. What is the final velocity ?

E’i = E’f + E’loss

K’i + U’i = K’f + U’f + E’loss

0 + mgh = ½mvf2 + 0 + fh

fmgf)(mg

vv

202

f

fmg

fmgvv 2

02f

mg = ½mvf2 + f f)2(mg

mv20

f)2(mg

mv20

½mvf2 = mg - f f)2(mg

mv20

f)2(mg

mv20

f)2(mg

mv20

Centre of Mass (1D)

0x1 x2

xcm

m1M m2

M = m1 + m2

M xcm = m1 x1 + m2 x2

Mxmxm

x 2211cm

In general iicm xmM1

x