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We have seen that work done by or against the gravitational force depends only on the starting and ending points, and not on the path between, allowing us to define the simplifying concept of gravitational potential energy. We can do the same thing for a few other forces, and we will see that this leads to a formal definition of the law of conservation of energy.

Making connections: take-home investigation—converting potential to kinetic energy

One can study the conversion of gravitational potential energy into kinetic energy in this experiment. On a smooth, level surface, use a ruler of the kind that has a groove running along its length and a book to make an incline (see [link] ). Place a marble at the 10-cm position on the ruler and let it roll down the ruler. When it hits the level surface, measure the time it takes to roll one meter. Now place the marble at the 20-cm and the 30-cm positions and again measure the times it takes to roll 1 m on the level surface. Find the velocity of the marble on the level surface for all three positions. Plot velocity squared versus the distance traveled by the marble. What is the shape of each plot? If the shape is a straight line, the plot shows that the marble’s kinetic energy at the bottom is proportional to its potential energy at the release point.

A book is lying on the table and one end of a ruler rests on the edge of this book while the other end rests on the table, making it an incline. A marble is shown rolling down the ruler.
A marble rolls down a ruler, and its speed on the level surface is measured.

Section summary

  • Work done against gravity in lifting an object becomes potential energy of the object-Earth system.
  • The change in gravitational potential energy, Δ PE g size 12{Δ"PE" rSub { size 8{g} } } {} , is ΔPE g = mgh size 12{Δ"PE" rSub { size 8{g} } = ital "mgh"} {} , with h size 12{h} {} being the increase in height and g size 12{g} {} the acceleration due to gravity.
  • The gravitational potential energy of an object near Earth’s surface is due to its position in the mass-Earth system. Only differences in gravitational potential energy, ΔPE g size 12{Δ"PE" rSub { size 8{g} } } {} , have physical significance.
  • As an object descends without friction, its gravitational potential energy changes into kinetic energy corresponding to increasing speed, so that ΔKE = − ΔPE g size 12{D"KE""=-"D"PE" rSub { size 8{g} } } {} .

Conceptual questions

In [link] , we calculated the final speed of a roller coaster that descended 20 m in height and had an initial speed of 5 m/s downhill. Suppose the roller coaster had had an initial speed of 5 m/s uphill instead, and it coasted uphill, stopped, and then rolled back down to a final point 20 m below the start. We would find in that case that it had the same final speed. Explain in terms of conservation of energy.

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Does the work you do on a book when you lift it onto a shelf depend on the path taken? On the time taken? On the height of the shelf? On the mass of the book?

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Problems&Exercises

A hydroelectric power facility (see [link] ) converts the gravitational potential energy of water behind a dam to electric energy. (a) What is the gravitational potential energy relative to the generators of a lake of volume 50 . 0 k m 3 size 12{"50" "." 0`"km" rSup { size 8{3} } } {} ( mass = 5 . 00 × 10 13 kg ) size 12{"mass"=5 "." "00" times "10" rSup { size 8{"13"} } `"kg " \) } {} , given that the lake has an average height of 40.0 m above the generators? (b) Compare this with the energy stored in a 9-megaton fusion bomb.

A dam with water flowing down its gates.
Hydroelectric facility (credit: Denis Belevich, Wikimedia Commons)

(a) 1 . 96 × 10 16 J size 12{1 "." "96" times "10" rSup { size 8{"16"} } " J"} {}

(b) The ratio of gravitational potential energy in the lake to the energy stored in the bomb is 0.52. That is, the energy stored in the lake is approximately half that in a 9-megaton fusion bomb.

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(a) How much gravitational potential energy (relative to the ground on which it is built) is stored in the Great Pyramid of Cheops, given that its mass is about 7  ×  10 9  kg size 12{7 times "10" rSup { size 8{9} } " kg"} {} and its center of mass is 36.5 m above the surrounding ground? (b) How does this energy compare with the daily food intake of a person?

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Suppose a 350-g kookaburra (a large kingfisher bird) picks up a 75-g snake and raises it 2.5 m from the ground to a branch. (a) How much work did the bird do on the snake? (b) How much work did it do to raise its own center of mass to the branch?

(a) 1.8 J

(b) 8.6 J

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In [link] , we found that the speed of a roller coaster that had descended 20.0 m was only slightly greater when it had an initial speed of 5.00 m/s than when it started from rest. This implies that Δ PE >> KE i size 12{D"PE>>KE" rSub { size 8{i} } } {} . Confirm this statement by taking the ratio of Δ PE size 12{D"PE"} {} to KE i size 12{"KE" rSub { size 8{i} } } {} . (Note that mass cancels.)

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A 100-g toy car is propelled by a compressed spring that starts it moving. The car follows the curved track in [link] . Show that the final speed of the toy car is 0.687 m/s if its initial speed is 2.00 m/s and it coasts up the frictionless slope, gaining 0.180 m in altitude.

A toy car is moving up a curved track.
A toy car moves up a sloped track. (credit: Leszek Leszczynski, Flickr)
v f = 2 gh + v 0 2 = 2 ( 9.80 m /s 2 ) ( 0 .180 m ) + ( 2 .00 m/s ) 2 = 0 .687 m/s size 12{v rSub { size 8{f} } = sqrt {2 ital "gh"+v rSub { size 8{0} rSup { size 8{2} } } } = sqrt {2 \( 9 "." "80"" m/s" rSup { size 8{2} } \) \( - 0 "." "180"" m" \) + \( 2 "." "00 m/s" \) rSup { size 8{2} } } =0 "." "687"" m/s"} {}
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In a downhill ski race, surprisingly, little advantage is gained by getting a running start. (This is because the initial kinetic energy is small compared with the gain in gravitational potential energy on even small hills.) To demonstrate this, find the final speed and the time taken for a skier who skies 70.0 m along a 30º size 12{"30"°} {} slope neglecting friction: (a) Starting from rest. (b) Starting with an initial speed of 2.50 m/s. (c) Does the answer surprise you? Discuss why it is still advantageous to get a running start in very competitive events.

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Questions & Answers

can some one tell me how v=RW is dimensionally correct?
YAGNAK Reply
ms-1 = m X Hz
babar
What is displacement
Megha Reply
shortest distance b/w two points
bilal
distance+direction
A.d
explain distanace+direction
bilal
the change of postion from one point to another with direction
A.d
if we change thrle direction then displacement is destroy?
bilal
change the direction then?
bilal
what do u mean by i didnt understand bro
A.d
displacement is one dimension...?
bilal
displacement is the total length an object cover from initial to the final with respect to direction as Well as time.
mohammed
thanks
bilal
displacement is the ratio of speed with respect to particular time
Bhautik
shortest distance travel from initial point to final point
ankit
is straight shortest line that connect initial pt with final pt.
Zeleke
what are the differences between vector and scalar quantity
Kabba Reply
vector is assigned to those physical quantity that has both direction and magnitude! example velocity ,scalar just has magnitude example Mass of an object. hope it helps
Mudang
velocity is produce in fan...?
bilal Reply
how many electrons are there in 5 microcouloumb
Obed Reply
can a given total amount of mechanical energy be totally converted into heat energy..if so give example
Muhammad Reply
human running
Emmanuel
what is the fumula for calculating specific heat capacity, fusion,fission and vaporization?
Dohn Reply
Q=cm(∆t)
Emmanuel
Q=cm∆T
Muhammad
what is difference b/w vaporization and evaporation
Muhammad
evaporation is the process of extracting moisture while vaporization is process of becoming a vapor or gas
Emmanuel
From a molecular standpoint they are both cooling processes. Also, you may want to explore states of matter😊 #myTwoCents ~Shi~
Shii
cooling is a similarlity in both process I am confused in difference
Muhammad
1- Evaporation is a process where a liquid change to gas without reaching its boiling point. 2- Vaporization is a process where a liquid change to gas after reaching its boiling point. 3- Sublimation is a process where a solid changes into vapour without passing through a liquid state
Victor
I see. Evaporation is a type of vaporization, that occurs on the surface of a liquid as it changes into the gaseous phase before reaching its boiling point. hope that aids
Shii
vaporisation is cooling process while vaporization is heating process
Emmanuel
I mean to write evaporation is an heating process while vaporization is cooling process
Emmanuel
Yea here are two applications. 1- your wet washed clothes dry under the sun, the water EVAPORATES 2- when u are cooking, it reaches a point where u need to add more water because the water you added previously is getting dried. this is VAPORIZATION. Am not sure which is a cooling or heating process
Victor
vaporization occur only when the evaporation get to level where the above cloud is been (saturated) so cooling take place and started to change to liquid (eg rain fall)
Emmanuel
They are both properties of the same process so they're both cooling
Shii
what about sublimation? cooling or heating process?
Victor
exact
Muhammad
evaporation is the increase in kinetic energy of the liquid which can be gone by adding heat
Emmanuel
so its an heating process
Emmanuel
sublimation is when a solid change to gas
Emmanuel
evaporation is very definitely a cooling process. respectfully@Emmanuel when liquid turns to gas it requires more energy from its surroundings, this energy is in the form of heat, and when heat energy leaves the evaporating liquid it leaves it cooler. Thus, cooling process.
Shii
.
Shii
evaporation is very definitely a cooling process. respectfully@Emmanuel
Shii
kk
Emmanuel
You're right @Shi. I get your point
Victor
eascape velocity on the surface of Earth is 11.2 kms-1 the escape velocity on the surface of another planet of same mass as that of Earth but of 1/4 times of radius of Earth is a5.6kms-1 b11.2 kms-1 c22.4kms-1 d5.6ms-1
Muhammad
Emm.. is that a question? or..
Victor
it is McQ
Muhammad
a)5.6km/s
Alvis
c= Q/cm◇T
A.d
it's answer is 22.4
Muhammad
units...
Shii
vital
Shii
the time period of the artificial satellite is given by ?
raza
Why is there no 2nd harmonic in the classical electron orbit?
Shree Reply
how to reform magnet after been demagneted
Inuwa Reply
A petrol engine has a output of 20 kilowatts and uses 4.5 kg of fuel for each hour of running. The energy given out when 1 kg of petrol is burnt is 4.8 × 10 to the power of 7 Joules. a) What is the energy output of the engine every hour? b) What is the energy input of the engine every hour?
Morris Reply
Issac Newton devised a genius way to calculate changing quantities...
Shii
what is the error during taking work done of a body..
Aliyu Reply
what kind of error do you think? and work is held by which force?
Daniela
I am now in this group
smart
theory,laws,principles and what-a-view are not defined. why? you
Douglas Reply
A simple pendulum is used in a physics laboratory experiment to obtain an experimental value for the gravitational acceleration, g . A student measures the length of the pendulum to be 0.510 meters, displaces it 10 o from the equilibrium position, and releases it. Using a s
Emmanuel Reply
so what question are you passing across... sir?
Olalekan
Two masses of 2 kg and 4 kg are held with a compressed spring between them. If the masses are released, the spring will push them away from each other. If the smaller mass moves off with a velocity of 6m/s, what is the stored energy in the spring when it is compressed?
Emmanuel Reply
54 joule
babar
how?
rakesh
Reduce that two body problem into one body problem. Apply potential and k. E formula to get total energy of the system
rakesh
i dont think dere is any potential energy... by d virtue of no height present
Olalekan
there is compressed energy,dats only potential energy na?
rakesh
yes.. but... how will u approach that question without The Height in the question?
Olalekan
Can you explain how you get 54J?
Emmanuel
Because mine is 36J
Emmanuel
got 36J too
Douglas
OK the answer is 54J Babar is correct
Emmanuel
Conservation of Momentum
Emmanuel
woow i see.. can you give the formula for this
joshua
Two masses of 2 kg and 4 kg are held with a compressed spring between them. If the masses are released, the spring will push them away from each other. If the smaller mass moves off with a velocity of 6m/s, what is the stored energy in the spring when it is compressed? Asume there is no external force.
Emmanuel Reply
Please help!
Emmanuel
please help find dy/dx 2x-y/x+y
Inuwa
By using the Quotient Rule dy/dx = 3y/(x +y)²
Emmanuel
3y/(x+y)²
Emmanuel
may be by using MC^2=MC^2 and Total energy=kinetic energy +potential energy so 1st find kinetic energy and den find potential energy which is stored energy
rakesh
i think i m correct
rakesh
But how?
Emmanuel
3y/(x+y)²
Douglas
Practice Key Terms 1

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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