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• Last Time:
Work, Kinetic Energy, Work‐Energy Theorem
• Today:
Gravitational Potential Energy, Conservation of Energy, Spring Potential Energy, Power
HW #4 extension to Friday 5:00 p mHW #4 extension to Friday, 5:00 p.m.
HW #5 il blHW #5 now available
Due Thursday, October 7, 11:59 p.m.
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Conceptual Question
A 10 N force acts on a block, as shown below (other forces may also b ti ) Th bl k th h i t l di t D i thbe acting). The block moves the same horizontal distance D in the +x‐direction in all four cases below. Rank the amount of work done by the 10 N force, in order of most positive, to most negative.by the 10 N force, in order of most positive, to most negative.
A B C D
θ
y
2
x
Conservative vs. Non‐Conservative Forces
In general, there are two kinds of forces :
“Conservative” Forces “Non‐Conservative” Forces
E b E bEnergy can be recovered
Energy cannot be recovered
E g Gravity E g FrictionE.g., Gravity E.g., Friction
3Generally: Dissipative
Gravitational Potential EnergySuppose an object falls from some height to a lower height.
How much work has been done by gravity ?How much work has been done by gravity ?
y0)0(cos|| ifg yymgW
|F| |Δ | θ
mg
|F| |Δy| cos θ
Δymgyi If an object is raised to some height, there is the “potential” for gravity to d iti k P iti kyf do positive work. Positive work means an increase in the object’s kinetic energy.
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kinetic energy.
Gravitational Potential EnergySo we then define the “gravitational potential energy”
Gravitational Potential Energy
PE = mgy
y: vertical position relative to Earth’s
f ( thPE = mgy surface (or another reference point)
SI unit: Joule
The gravitational potential energy quantifies the magnitude of work that can be done by gravity.
By the Work‐Energy Theorem, the gravitational potential energy is then equal to the change in the object’s kinetic energy if it falls di
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a distance y.
Reference Level for Potential EnergyWe have defined the gravitational potential energy to be:
mgyPE
Q: Does it matter where we define y = 0 to be ?
A: No, it doesn’t matter. All that matters is the difference in the potential energy, ΔPE = mg Δy . It doesn’t matter where we define zero to bedefine zero to be.
100 m 5 mIn both of these, the object falls 5 m.
695 m 0 m
object falls 5 m.
Gravity and Conservation of EnergyConservation Law : If a physical quantity is “conserved”, the numerical value of the physical quantity remains unchangednumerical value of the physical quantity remains unchanged.
PEKEPEKE
Conservation of Mechanical Energy :
ffii PEKEPEKE
Sum of kinetic energy and gravitational potential energy remains constant at all times. It is a conserved quantity.
If we denote the total mechanical energy as E = KE + PE, the total mechanical energy E is conserved at all times
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total mechanical energy E is conserved at all times.
Gravity and Conservation of EnergyIgnoring dissipative forces (air resistance), at all times the total mechanical energy will be conserved :mechanical energy will be conserved :
112
221
21 2
121 mgymvmgymv
initial total final totalinitial total mechanical energy
final total mechanical energy
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ExampleA 25‐kg object is dropped from a height of 15.0 m above the ground Assuming air resistance is negligibleground. Assuming air resistance is negligible …
(a) What is its speed 7 0 m above the ground ?(a) What is its speed 7.0 m above the ground ?
(b) What is its speed when it hits the ground ?( ) p g
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ExampleA skier starts from rest at the top of a frictionless ramp of height 20 0 m At the bottom of the ramp the skier encounters a20.0‐m. At the bottom of the ramp, the skier encounters a horizontal surface where the coefficient of kinetic friction is μk = 0.210. Neglect air resistance.μk 0.210. Neglect air resistance.
(a) Find the skier’s speed at the bottom of the ramp.( ) p p
(b) How far does the skier travel on the horizontal surface before coming to rest ?
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Pendulum and Conservation of Energy
AC
B
A pendulum is released from rest at point A. Ignoring friction …
What is its speed at the bottom of its trajectory at B ?What is its speed at the bottom of its trajectory at B ?
How high does it swing on its way up to C ?
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When it swings back to A, does it return to its initial height ?
Springs
One must do work on a spring to compress or stretch itOne must do work on a spring to compress or stretch it.
The work it takes to compress or stretch the spring can be recovered as kinetic energy.
This means we can find a potential energy function for springs, which we can then use in the Work‐Energy Theorem.
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Hooke’s Law
x 0 Position of spring when notx = 0 : Position of spring when not compressed/stretched
x
compressedx = 0 : Force exerted by spring :
F = –kx k: “spring constant”
x
p g
If compressed, x < 0, so F > 0 !
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x
Hooke’s Law
x 0 Position of spring when notx = 0 : Position of spring when not compressed/stretched
x
stretchedx
x = 0 : Force exerted by spring :
F = –kxx
If stretched, x > 0, so F < 0 !
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Spring Potential Energy
compressedx = 0
stretchedx = 0 :
compressed stretchedx
x
If spring is compressed or stretched, it will exert a force, and so it has the potential to do workhas the potential to do work.
Elastic potential energy associated with this spring force is :
21 kxPEs k : “spring constant”
x : displacement of spring
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2kxPEs x : displacement of spring
SI Unit: Joules
Springs and Conservation of Energy
Assuming only conservative forces (i.e., no non‐conservative g y ( ,forces, such as friction), systems with springs will obey :
fgsigs PEPEKEPEPEKE
Initial Initial Initial Final Final FinalInitial KE
Initial Spring PE
Initial Grav. PE
Final KE
Final Spring PE
Final Grav. PE
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Non‐Conservative Forces
If there is a non‐conservative force (e.g., friction) acting, work d b h fdone by this force is :
)()( PEKEPEKEW Mechanical energy )()( iiffnc PEKEPEKEW gychanges. Work done by non‐conservative force dissipated as, e.g., heat.if EE if
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Example 5.9 (p. 137)
A block with mass of 5.0 kg is attached to a horizontal spring with g p gspring constant k = 400 N/m. The surface the block rests on is frictionless. If the block is pulled out to xi = 0.05 m and released …
(a) Find the speed of the block when it reaches the equilibrium point (x = 0).
(b) Find the speed when x = 0.025 m.
(c) Repeat (a) if friction acts, with μk = 0 150
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with μk 0.150.
Power
20 Watt CFL Light Bulb
300 hp Ford Mustang
If l f d k W bj i i l ΔIf an external force does work W on an object in some interval Δt, then the average power delivered to the object is the work done divided by the time intervaldivided by the time interval
WP W in JoulesΔt in seconds
tP
Δt in seconds
P in Watts = Joule/second
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The higher the power, the more work that can be done in a given time interval.
Power
Note that we can write :
vFtxF
txF
tWP
ttt
A i t t f ti th dAverage power is a constant force times the average speed.
Note on units :
1 Watt = 1 Joule/second = 1 kg‐m2/s3
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Example: 5.53The electric motor of a model train accelerates the train from rest of 0 620 m/s in 0 021 s The total mass of the train is 0 875 kgof 0.620 m/s in 0.021 s. The total mass of the train is 0.875 kg. Find the average power delivered to the train during its acceleration.acceleration.
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Next Class
• 6.1 – 6.2 :
Momentum, Impulse, Conservation of Momentum
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