Physics 1408-002 Principles of Physics Lecture 13 – Chapter 8 – February 23, 2009 Sung-Won Lee [email protected]
Announcement II SI session by Reginald Tuvilla SI sessions will be at the following times and location. Monday 4:30 - 6:00pm - Holden Hall 106 Thursday 4:00 - 5:30pm - Holden Hall 106
Lecture note is on the web Handout (6 slides/page) http://highenergy.phys.ttu.edu/~slee/1408/ *** Class attendance is strongly encouraged and will be taken randomly. Also it will be used for extra credits.
HW Assignment #5 will be placed on MateringPHYSICS today, and is due by 11:59pm on Wendseday, 2/25
Chapter 8 Conservation of Energy •! Conservative and Non-conservative Forces •! Potential Energy •! Mechanical Energy and Its Conservation •! Problem Solving Using Conservation of Mechanical Energy •! The Law of Conservation of Energy •! Energy Conservation with Dissipative Forces: Solving Problems •! Gravitational Potential Energy and Escape Velocity •! Power
8-1 Conservative and Nonconservative Forces A force is conservative if: the work done by the force on an object moving from one point to another depends only on the initial and final positions of the object, and is independent of the particular path taken. Example: gravity.
Object of mass m: (a) falls a height h vertically; (b) is raised along an arbitrary two-dimensional path.
8-2 Potential Energy (P.E.) •! We have now seen a method of storing energy in a system - kinetic energy. The 2nd method, which we cover in Chapter 8, is potential energy. •! Potential energy is energy related to the configuration of a system in which the components of the system interact by forces. •! Examples include: –!elastic potential energy – stored in a spring –!gravitational potential energy –!electrical potential energy
8-2 Potential Energy
8-2 Potential Energy
In raising a mass m to a height h, the work done by the external force is
. WG = FG d =
A person exerts an upward force Fext = mg to lift a brick from y1 to y2 .
General definition of gravitational potential energy: WG = -mgh
For any conservative force:
We therefore define the gravitational potential energy at a height y above . some reference point:
8-2 Potential Energy A spring has potential energy, called elastic potential energy, when it is compressed. The force required to compress or stretch a spring is:
8-2 Potential Energy Then the potential energy is:
where k is called the spring constant, and needs to be measured for each spring. A spring (a) can store energy (elastic potential energy) when compressed (b), which can be used to do work when released (c) and (d).
8-2 Potential Energy In one dimension,
Conservation of Mechanical Energy •! Look at work done by the book as it falls from some height to a lower height •! From work-kinetic energy theorem of Ch.7, Won book = !Kbook
We can invert this equation to find U(x) if we know F(x):
•! Also, W = F!r = mgyb – mgya = !Kbook •! mgyb – mgya = -(Uf - Ui) = -!Ug •! So, !K = -!Ug => !K + !Ug = 0 "
In three dimensions:
•! We define the sum of kinetic and potential energies as mechanical energy in the system: Emech = K + Ug •! The statement of Conservation of Mechanical Energy for an isolated system: Kf + Uf = Ki+ Ui
8-3 Mechanical Energy and Its Conservation If there are no nonconservative forces, the sum of the changes in the kinetic energy and in the potential energy is zero—the kinetic and potential energy changes are equal but opposite in sign.
8-4 Problem Solving Using Conservation of Mechanical Energy In the image on the left, the total mechanical energy at any point is:
This allows us to define the total mechanical energy:
And its conservation:
8-4 Problem Solving Using Conservation of Mechanical Energy If the original height of the rock is y1 = h = 3.0 m, calculate the rock’s speed when it has fallen to 1.0 m above the ground.
8-4 Problem Solving Using Conservation of Mechanical Energy Assuming the height of the hill is 40 m, and the roller-coaster car starts from rest at the top, calculate (b) at what height it will have half this speed. Take y = 0 at the bottom of the hill.
The rock’s potential energy changes to kinetic energy as it falls. Note bar graphs representing potential energy U and kinetic energy K for the three different positions.
8-4 Problem Solving Using Conservation of Mechanical Energy Assuming the height of the hill is 40 m, and the roller-coaster car starts from rest at the top, calculate (a) the speed of the roller-coaster car at the bottom of the hill
8-4 Problem Solving Using Conservation of Mechanical Energy Estimate the kinetic energy and the speed required for a 70-kg pole vaulter to just pass over a bar 5.0 m high. Assume the vaulter’s center of mass is initially 0.90 m off the ground and reaches its maximum height at the level of the bar itself.
8-4 Problem Solving Using Conservation of Mechanical Energy For an elastic force, conservation of energy tells us:
A dart of mass 0.100 kg is pressed against the spring of a toy dart gun. The spring (with spring constant k = 250 N/m and ignorable mass) is compressed 6.0 cm and released. If the dart detaches from the spring when the spring reaches its natural length (x = 0), what speed does the dart acquire?
8-4 Problem Solving Using Conservation of Mechanical Energy Example 8-8: Two kinds of potential energy A ball of mass m = 2.60 kg, starting from rest, falls a vertical distance h = 55.0 cm before striking a vertical coiled spring, which it compresses an amount Y = 15.0 cm.
8-4 Problem Solving Using Conservation of Mechanical Energy A dart of mass 0.100 kg is pressed against the spring of a toy dart gun. The spring (with spring constant k = 250 N/m and ignorable mass) is compressed 6.0 cm and released. If the dart detaches from the spring when the spring reaches its natural length (x = 0), what speed does the dart acquire?
A re-look at some problems Let’s say that we want to know the velocity of a block sliding on a frictionless inclined plane after it has slid down from a height h.
We determined the acceleration down the plane before using F = ma
Determine the spring constant. Assume the spring has negligible mass, and ignore air resistance.
adown plane = g sin! 2
v2 = v2 + 2 a s = v + 2 g sin! (h / sin!) 0
v2 = v2 + 2 g h 0
v = "2 g h
if it starts from rest
Using Energy Conservation Let’s say that we want to know the velocity of a block sliding on a frictionless inclined plane after it has slid down from a height h.
Kf + Uf = Ki + Ui 2 1 – mv 2 f
1 – 2
Here the acceleration down the plane is continually changing since the angle of plane with the horizontal changes.
mv2 + mgh
vf = "2 g h
Kf + Uf = Ki + Ui
vf2 = vi 2 + 2 g h if it starts from rest
Now what about a block sliding down an incline like this
2 1 – mv f 2 2
+ 0 =1– mvi 2 + mgh 2
vf = vi + 2 g h if it starts from rest
vf = "2 g h
A Swinging Pendulum starts from rest at height h
Total Energy is same everywhere.
solve for velocity at bottom of swing
Kf + Uf = Ki + Ui 2 + 0 =1– mv + mgh
2 1 – mv f 2 2
v f = vi + 2 g h
v f= "2 g h
8-5 The Law of Conservation of Energy Nonconservative forces: Friction Heat Electrical energy Chemical energy and more do not conserve mechanical energy. However, when these forces are taken into account, the total energy is still conserved:
What About Friction? Work done by friction is always
since Ffr is always opposite displacement
Wfr = – Ffr l = µN l where “l” is the total path length (displacement)
Top of swing: v=0, K=0 (A,C) all U
v = "2 g h
Bottom of swing: U=0 (B) max v, All K
8-5 The Law of Conservation of Energy The law of conservation of energy is one of the most important principles in physics.
The total energy is neither increased nor decreased in any process. Energy can be transformed from one form to another, and transferred from one object to another, but the total amount remains constant.
When friction is acting, mechanical energy is not conserved
#K = Wtot = $ Wcons + $ Wnon-cons = – $ ( #U) + $ Wnon-cons #K = – $ ( #U) – Ffrl Kf + $ Uf = Ki + $ Ui – Ffrl Mechanical energy is not conserved due to friction force
Kf + $ Uf < Ki + $ Ui
Where did it go ?
Block down a plane with friction Let it start from rest
Kf + Uf = Ki + Ui – Ffrs
2 1 – mv f 2
!! 2 1 – mv f 2
= mgh – Ffrs
= mgh – (µ m g cos!) s
vf2 = 2 g h – 2 µg cos! (h / sin! )
vf = "2 g h (1 – µ cot !) e.g. no friction force; vf = "2 g h
8-6 Energy Conservation with Dissipative Forces: Solving Problems Determine the thermal energy produced and estimate the average friction force (assume it is roughly constant) on the car, whose mass is 1000 kg.
8-6 Energy Conservation with Dissipative Forces: Solving Problems Example: Friction on the roller-coaster car. The roller-coaster car shown reaches a vertical height of only 25 m on the second hill before coming to a momentary stop. It traveled a total distance of 400 m. Determine the thermal energy produced and estimate the average friction force (assume it is roughly constant) on the car, whose mass is 1000 kg.
8-6 Energy Conservation with Dissipative Forces: Solving Problems Example: Friction with a spring. A block of mass m sliding along a rough horizontal surface is traveling at a speed v0 when it strikes a massless spring headon and compresses the spring a maximum distance X. If the spring has stiffness constant k, determine the coefficient of kinetic friction between block and surface.
Because of friction, a roller-coaster car does not reach the original height on the second hill. The difference in the initial and final energies is the thermal energy produced, 147000 J. This is equal to the average frictional force multiplied by the distance traveled, so the average force is 370 N.
8-7 Gravitational Potential Energy and Escape Velocity Far from the surface of the Earth, the force of gravity is not constant:
The work done on an object moving in the Earth’s gravitational field is given by:
Arbitrary path of particle of mass m moving from point 1 to point 2.
8-7 Gravitational P.E. and Escape Velocity Solving the integral gives:
We can define gravitational potential energy:
8-7 Gravitational Potential Energy and Escape Velocity Example: Package dropped from high-speed rocket. A box of empty film canisters is allowed to fall from a rocket traveling outward from Earth at a speed of 1800 m/s when 1600 km above the Earth’s surface. The package eventually falls to the Earth. Estimate its speed just before impact. Ignore air resistance.
8.8 Power •! The time rate of energy transfer •! The average power is given by
Instantaneous Power •! The instantaneous power is the limiting value of the average power as #t approaches zero
•! This can also be written as
Summary of Chapter 8 •!Gravitational potential energy: Ugrav = mgy. •!Elastic potential energy: Uel = ! kx2. •!For any conservative force:
•!Total mechanical energy is the sum of kinetic and potential energies. •!Additional types of energy are involved when nonconservative forces act. •!Gravitational potential energy: •!Power:
8-7 Gravitational Potential Energy and Escape Velocity If an object’s initial kinetic energy is equal to the potential energy at the Earth’s surface, its total energy will be zero. The velocity at which this is true is called the escape velocity; for Earth:
Power Generalized •! Power can be related to any type of energy transfer •! In general, power can be expressed as
Units of Power •! The SI unit of power is called the [watt] –! 1 watt = 1 joule / second = 1 kg . m2 / s2
•! US Customary system is horsepower: 1 hp = 746 W •! Unit of energy can be defined in terms of units of power. –! 1 kWh (kilowatt-hour)= (1000 W)(3600 s) = 3.6 x106 J