Download Lecture 10-2

Welcome back to Physics 215
Today s agenda:
• Work
• Power
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Class next Tuesday in a different location!
• Physics 202 (we’ll have some extra guests)
• Reading for next week:
– Lecture 11-­1: Extended Objects (12.1-­
12.3)
– Lecture 11-­2: Torque (12.4-­12.5)
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SG A person carries a book horizontally at constant speed. The work done on the book by the person s hand is
A. positive
B. negative
C. equal to zero
D. Can t tell.
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Sample problem: The two ropes seen in the figure are used to lower a 255 kg piano 5.1 m from a second-­story window to the ground. How much work is done by each of the three forces?
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SG Two identical blocks slide down two frictionless ramps. Both blocks start from the same height, but block A is on a steeper incline than block B. Using the work-­kinetic energy theorem, the speed of block A at the bottom of its ramp is A. less than the speed of block B. B. equal to the speed of block B.
C. greater than the speed of block B.
D. Can t tell.
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Solution • Which forces do work on block?
• Which, if any, are constant?
• What is F•Ds for motion?
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Work done by gravity
Work W = -mg j•Ds
N
Ds
Therefore,
W = -­mgDh
mg
j
N does no work!
i
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Work done on an object by gravity
W (on object by earth) = – m g Dh, where Dh = hfinal – hinitial is the change in height.
Object moves
Change in
height Δh
Work done
on object
by earth
upward
positive
negative
downward
negative
positive
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Defining gravitational potential energy
W(on obj. by earth) = ΔK
0 = ΔK − W(on obj. by earth)
0 = ΔK + ΔUg
The change in gravitational potential energy of the object-­
earth system is just another name for the negative value of the work done on an object by the earth.
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Curved ramp
Ds =
W = F•Ds =
Work done by gravity between 2
fixed pts does not depend on path
taken!
Work done by gravitational force in moving some
object along any path is independent of the path
depending only on the change in vertical height
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Demo: 4 tracks
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Conservative forces
• If the work done by some force (e.g. gravity) does not depend on path the force is called conservative.
• Then gravitational potential energy Ug
only depends on (vertical) position of object Ug = Ug(h)
• Elastic forces also conservative –
elastic potential energy U = (1/2)kx2...
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Many forces
• For a particle which is subject to several (conservative) forces F1, F2 …
DK= W1+W2+… or equivalently:
DK+DU1+DU2+….=0 i.e E = (1/2)mv2 + U1 + U2 +… is constant
• Principle called Conservation of total mechanical energy
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Nonconservative forces
• Can do work, but cannot be represented by a potential energy function
– examples: friction, air resistance
• Total mechanical energy can now change due to work done by nonconservative force
DK+DU1+DU2+….= Wnonconservative
• For example, frictional force leads to decrease of total mechanical energy -­-­ energy converted to heat, or internal energy • Heat/internal energy is really just energy of molecular motion
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Demo: work done by gravity as measured by soda cans
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A 5.00-kg package slides 1.50 m down a long ramp that is inclined at 12.0° below the
horizontal. The coefficient of kinetic friction between the package and the ramp is µk =
0.310. Calculate: (a) the work done on the package by friction; (b) the work done on
the package by gravity; (c) the work done on the package by the normal force; (d) the
total work done on the package. (e) If the package has a speed of 2.20 m/s at the top of
the ramp, what is its speed after sliding 1.50 m down the ramp?
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Conservation of total energy
The total energy of an object or system is said to be conserved if the sum of all energies (including those outside of mechanics that have not yet been discussed) never changes.
This is believed always to be true.
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Summary
• Work is defined as dot product of force with displacement vector
• Each individual force on an object gives rise to work done
• The kinetic energy only changes if net work is done on the object, which requires a net force
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Power
Power = Rate at which work is done
Average power =
W
Δt
W
Δt →0 Δt
Inst. power = lim
Units of power:
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SG A sports car accelerates from zero to 30 mph in 1.5 s. How long does it take to accelerate from zero to 60 mph, assuming the power (=DW / Dt) of the engine to be constant?
(Neglect losses due to friction and air drag.)
A.
B.
C.
D.
2.25 s
3.0 s
4.5 s
6.0 s
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Power in terms of force and velocity
 
W F ⋅ Δs
Power = =
Δt
Δt
 Δs  
=F⋅
= F⋅v
Δt
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SG A locomotive accelerates a train from rest to a final speed of 40 mph by delivering constant power. If we assume that there are no losses due to air drag or friction, the acceleration of the train (while it is speeding up) is
A.
B.
C.
decreasing
constant
increasing
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SG A ball is whirled around a horizontal circle at constant speed. If air drag forces can be neglected, the power expended by the hand is:
A.
B.
C.
D.
positive
negative
zero
Can t tell.
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Summary: Power
• Power:
– P = dW/dt
– P = F v cos q
• It’s the rate at which work or energy is transferred or transformed
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Rigid Body rotation
Knight 4th edition Chapter 12
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Motion of Real Objects
• So far discussed motion of idealized point-­like objects
• Saw that neglecting internal forces OK
– only net external forces need to be considered for translational motion of center of mass
– what about rotational motion?
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Rigid Bodies
• Real extended objects can move in complicated ways (stretch, twist, etc.)
• Here, think of relative positions of each piece of body as fixed – idealize object as rigid body
• Can still undergo complicated motion (translational motion plus rotations)
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Center of Mass
• Properties:
– When a collection of particles making up an extended body is acted on by external forces, the center of mass moves as if all the mass of the body were concentrated there.
– weight force can be considered to act vertically through center of mass
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Center of mass
for system of (point) objects:



(m1r1 + m2 r2 +…)
rCM =
(m1 + m2 +…)

1
= ∑ mi ri
M i
where M = (m1 + m2 +…)
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Points to note
• All real bodies are just collections of point-­like objects (atoms)
• It is not necessary that CM lie within volume of body
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Center of mass board demos
• How can we find center of mass for funny-­shaped object?
• Suspend from 2 points and draw in plumb lines.
• Where lines intersect yields CM
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