Work and Kinetic Energy (ch 6)!

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Work with Constant Forces!

Definition of Work: (1D, constant F, straight-line displacement)!A355"'*%.%!"#$%&#%'(")!*'B%.02'%2,#"34,%.%+,$%&#!*%(%.6")4%

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Work done?!A child is swinging a ball on the end of a piece of string round their head making it move in a perfect circle at a constant speed. If the tension in the string is 10N and the distance the ball moves during one complete circle is 2m how much work is done by the string per circuit?!

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What if the child swings this with changing

speed?

Developing an intuition for this definition:!

1) Larger forces (for fixed displacements) correspond to more work done.!

2) Larger displacements (for fixed forces) correspond to more work done.!

3) If the displacement is zero, W=0.!! (that car that you try to push with all your might but doesn’t move,!! !my friend, you have done NO work) !

4) If the force and the displacement are perpendicular, W=0. (Thus “centripetal forces” [net forces] do NO work.)!

5) Work is a scalar quantity.!

6) If 90°< " " 180° then the work done by the force will be Negative. (E.g. frictional forces opposing motion, work you did in lowering a box)!

7) F*cos(") is the component of F along s. Or, alternatively s*cos(") is the component of s along F.!

When more than one force acts on an object, the total work is the sum of the work done by each force separately.!

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It is the total work that enters in the work-energy theorem, the central result of this chapter.!

Total Work!

Example: Early one October, you go to a pumpkin patch to select your favorite Halloween pumpkin. You lift the 3.0 kg pumpkin to a height of 1.5 m at a constant speed, then carry it 50.0 m on level ground to the check-out stand.!(a) Calculate the work you do on the pumpkin as you lift it from the ground?!(b) What is the work done on the pumpkin by gravity as you lift it from the ground?!(c) What is the total work done on the pumpkin as it is lifted from the ground? !(d) How much work do you do on the pumpkin as you carry it from the field?!

(a)! Constant velocity --" weight = lift force! Wyou = m g h = 3.0 x 9.81 x 1.5 = 44.15 J!

Answers!

(b) weight = lift force, but weight is down (180 deg) from displacement. Wg = mg h cos180

o = -44.15 J!

(c) Net force in y = 0, total work = 0!

(d) If assume constant waking velocity, 0!

Example!

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Solution 1: -'

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Solution 2: Use vector dot product !

First write the pulling force and the displacement in!vector notation taking right and up as positive!

!

!

Work =! p "! s

!

= (pxˆ x + py

ˆ y ) " (100 ˆ x + 0 ˆ y )

!

= (pcos60 ˆ x + psin60 ˆ y ) " (100 ˆ x + 0 ˆ y )

!

= pcos60*100 + 0

Recognizing that the horizontal component of force F balances the friction force.!

!

pcos60 = f s = µmg = 0.3"130 "10 = 390N

!

Work = µmg"100 = 39000 J

!

W = Fd = F("x) = F(x2# x

1)

Constant force in 1D!

Variable (position-dependent) force

Geometrical

Interpretation

!

W = Fa"xa + Fb"xb + ... + Ff"x f

as

Example 1: Hooke’s law – linear

springs

Consider the work we do to slowly stretch or

compress a linear spring a distance x:

Method 1:

Method 2:

Note: The work the spring does is . The negative sign

indicates the opposite direction of the force relative to the displacement.

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The Work-Energy Theorem!

Toy Version (Fnet constant, 1D, straight-line displacement)!

!

Wtotal

= Fnet("x) = ma("x)

(const a!)

A More General Work-Energy Theorem!

Consider a NET position-dependent (i.e. variable) force Fnet =F(x)

in 1-dimension:

(Only if we are considering Fnet

can we make this step!)

! = f

i

x

x dx

dt

dv m

dx

dt

dx

dx

dv m

f

i

x

x ! =

! = f

i

x

x dx

dx

dv mv ! =

f

i

v

v dv mv

Same

result as toy version!

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2

1

2

1

i f mv mv ! =

!

= mv f2 " vi

2

2(#x)

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% &

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( ) (#x) =

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2mvi

2

Kinetic Energy!

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Conservation vs. Constancy!

!! It will slow down over time because energy is lost from the system due to the work done by friction between the curling rock and the ice

! !

Nope, not these conservative forces…!

Conservative Forces!

!!Forces can be classified into one of two groups !! Conservative or non-conservative

!! A conservative force acting on the object will have done

no net work when the object returns to its start location

!! A non-conservative force depends on the path an object takes rather than the object's position

Conservative Forces!(1) Have potential energy associated with them (non-conservative forces do not!) !

(2) Are not path dependent (non-conservative forces are path dependent!) !

Example of Conservative forces (real ones):

Example of Conservative forces (real ones): Friction (Drag)

Gravity EM Force Spring Force

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Gravitational Potential Energy!

Work Done = mgh m Height= h

Gravitational Potential Energy= mgh

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!! i.e. It is the same for all heights so if an object comes back to the same height as it starts at the gravitational force has done no work

Nuggets Regarding Potential Energy