Download Work done?

Work and Kinetic Energy (ch 6)!
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Work with Constant Forces!
Definition of Work:
displacement)!
(1D, constant F, straight-line
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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.
“centripetal forces” [net forces] do NO work.)!
5)
(Thus
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.
is the component of s along F.!
Or, alternatively
s*cos(")
Total Work!
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.!
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?!
Answers!
(a)! Constant velocity --" weight = lift force!
Wyou = m g h = 3.0 x 9.81 x 1.5 = 44.15 J!
(b) weight = lift force, but weight is down (180 deg) from
displacement. Wg = mg h cos180o = -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
= ( px xˆ + py yˆ ) " (100 xˆ + 0 yˆ )
= ( pcos60 xˆ + psin60 yˆ ) " (100 xˆ + 0 yˆ )
! = 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
!
!
Geometrical
Interpretation
Constant force in 1D!
W = Fd = F("x) = F(x 2 # x1 )
Variable (position-dependent)
force
!
W = Fa "x a + Fb "x b + ... + F f "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.
Energy!
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The Work-Energy Theorem!
Toy Version (Fnet constant, 1D, straight-line displacement)!
W total
$ v 2f " v i2 '
1 2 1 2
)(#x) = mv f " mv i
= Fnet ("x) = ma("x) = m&
2
2
% 2(#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!)
dv
dx
xi
dt
x
dv dx
=! m
dx
x
dx dt
xf
=! m
f
i
=!
xf
xi
dv
mv
dx
dx
=
vf
!v
Same
mv dv =
i
Kinetic Energy!
1 2 1 2
mv f ! mvi result as
2
2
toy version!
Conservation vs. Constancy!
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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
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Conservative Forces!
Nope, not these conservative forces…!
!
!
Forces can be classified into one of two groups
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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):
Gravity
EM Force
Spring Force
Example of Conservative forces (real ones): Friction (Drag)
Gravitational Potential Energy!
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Work Done = mgh
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Height= h
m
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Gravitational Potential Energy= mgh
Nuggets Regarding Potential Energy
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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
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