# 28.6 Relativistic energy  (Page 3/12)

 Page 3 / 12

## Calculating rest mass: a small mass increase due to energy input

A car battery is rated to be able to move 600 ampere-hours $\left(A·h\right)$ of charge at 12.0 V. (a) Calculate the increase in rest mass of such a battery when it is taken from being fully depleted to being fully charged. (b) What percent increase is this, given the battery’s mass is 20.0 kg?

Strategy

In part (a), we first must find the energy stored in the battery, which equals what the battery can supply in the form of electrical potential energy. Since ${\text{PE}}_{\text{elec}}=\text{qV}$ , we have to calculate the charge $q$ in $600\phantom{\rule{0.25em}{0ex}}A·h$ , which is the product of the current $I$ and the time $t$ . We then multiply the result by 12.0 V. We can then calculate the battery’s increase in mass using $\Delta E={\text{PE}}_{\text{elec}}=\left(\Delta m\right){c}^{2}$ . Part (b) is a simple ratio converted to a percentage.

Solution for (a)

1. Identify the knowns. $I\cdot t=\text{600 A}\cdot \text{h}$ ; $V=\text{12}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{V}$ ; $c=3\text{.}\text{00}×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m/s}$
2. Identify the unknown. $\Delta m$
3. Choose the appropriate equation. ${\text{PE}}_{\text{elec}}=\left(\Delta m\right){c}^{2}$
4. Rearrange the equation to solve for the unknown. $\Delta m=\frac{{\text{PE}}_{\text{elec}}}{{c}^{2}}$
5. Plug the knowns into the equation.
$\begin{array}{lll}\Delta m& =& \frac{{\text{PE}}_{\text{elec}}}{{c}^{2}}\\ & =& \frac{\text{qV}}{{c}^{2}}\\ & =& \frac{\left(It\right)V}{{c}^{2}}\\ & =& \frac{\left(\text{600 A}\cdot \text{h}\right)\left(\text{12.0 V}\right)}{\left(3.00×{\text{10}}^{8}{\right)}^{2}}\text{.}\end{array}$

Write amperes A as coulombs per second (C/s), and convert hours to seconds.

$\begin{array}{lll}\Delta m& =& \frac{\left(\text{600 C/s}\cdot \text{h}\left(\frac{\text{3600 s}}{\text{1 h}}\right)\left(\text{12.0 J/C}\right)}{\left(3.00×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m/s}{\right)}^{2}}\\ & =& \frac{\left(2.16×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{C}\right)\left(\text{12.0 J/C}\right)}{\left(3.00×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m/s}{\right)}^{2}}\end{array}$

Using the conversion $1\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot {\text{m}}^{2}{\text{/s}}^{2}=1\phantom{\rule{0.25em}{0ex}}\text{J}$ , we can write the mass as

$\Delta m=2.88×{\text{10}}^{-\text{10}}\phantom{\rule{0.25em}{0ex}}\text{kg}\text{.}$

Solution for (b)

1. Identify the knowns. $\Delta m=2.88×{\text{10}}^{-\text{10}}\phantom{\rule{0.25em}{0ex}}\text{kg}$ ; $m=\text{20.0 kg}$
2. Identify the unknown. % change
3. Choose the appropriate equation. $\text{}\text{% increase}=\frac{\Delta m}{m}×\text{100%}\text{}$
4. Plug the knowns into the equation.
$\begin{array}{lll}\text{% increase}& =& \frac{\Delta m}{m}×\text{100%}\\ & =& \frac{2.88×{\text{10}}^{-\text{10}}\phantom{\rule{0.25em}{0ex}}\text{kg}}{\text{20.0 kg}}×\text{100%}\\ & =& 1.44×{\text{10}}^{-9}\text{%}\text{.}\end{array}$

Discussion

Both the actual increase in mass and the percent increase are very small, since energy is divided by ${c}^{2}$ , a very large number. We would have to be able to measure the mass of the battery to a precision of a billionth of a percent, or 1 part in ${10}^{11}$ , to notice this increase. It is no wonder that the mass variation is not readily observed. In fact, this change in mass is so small that we may question how you could verify it is real. The answer is found in nuclear processes in which the percentage of mass destroyed is large enough to be measured. The mass of the fuel of a nuclear reactor, for example, is measurably smaller when its energy has been used. In that case, stored energy has been released (converted mostly to heat and electricity) and the rest mass has decreased. This is also the case when you use the energy stored in a battery, except that the stored energy is much greater in nuclear processes, making the change in mass measurable in practice as well as in theory.

## Kinetic energy and the ultimate speed limit

Kinetic energy is energy of motion. Classically, kinetic energy has the familiar expression $\frac{1}{2}{\mathrm{mv}}^{2}$ . The relativistic expression for kinetic energy is obtained from the work-energy theorem. This theorem states that the net work on a system goes into kinetic energy. If our system starts from rest, then the work-energy theorem is

${W}_{\text{net}}=\text{KE}.$

Relativistically, at rest we have rest energy ${E}_{0}={\mathrm{mc}}^{2}$ . The work increases this to the total energy $E={\mathrm{\gamma mc}}^{2}$ . Thus,

#### Questions & Answers

hi guys can you teach me how to solve a logarithm?
Villaflor Reply
how about a conceptual framework can you simplify for me? needed please
Villaflor
Hello what happens when electrone stops its rotation around its nucleus if it possible how
Afzal
I think they are constantly moving
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yep what is problem you are stuck into context?
S.M
not possible to fix electron position in space,
S.M
Physics
Beatriz
yes of course Villa flor
David
equations of kinematics for constant acceleration
Sagcurse Reply
A bottle full of water weighs 45g when full of mercury,it weighs 360g.if the empty bottle weighs 20g.calculate the relative density of mercury and the density of mercury....pls I need help
Lila Reply
well You know the density of water is 1000kg/m^3.And formula for density is density=mass/volume Then we must calculate volume of bottle and mass of mercury: Volume of bottle is (45-20)/1000000=1/40000 mass of mercury is:(360-20)/1000 kg density of mercury:(340/1000):1/50000=(340•40000):1000=13600
Sobirjon
the latter is true
Sobirjon
100g of water is mixed with 60g of a liquid of relative density 1.2.assuming no changes in volume occurred,find the average relative density of the mixture...take density of water as 1g/cm3 and density of liquid 1.2g/cm3
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plz hu can explain Heisenberg's uncertainty principle
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who can help me with my problem about acceleration?
Vann Reply
ok
Nicholas
how to solve this... a car is heading north then smoothly made a westward turn during the travel the speed of the car remains constant at 1.5km/h what is the acceleration of the car? the total travel time of the car as it smoothly changed its direction is 15 minutes
Vann
i think the acceleration is 0 since the car does not change its speed unless there are other conditions
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yes I have to agree, the key phrase is, "the speed of the car remains constant...," all other information is not needed to conclude that acceleration remains at 0 during the entire time
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who can help me with a relative density question
Lila
1cm3 sample of tin lead alloy has mass 8.5g.the relative density of tin is 7.3 and that of lead is 11.3.calculate the percentage by weight of tin in the alloy. assuming that there is no change of volume when the metals formed the alloy
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morning, what will happen to the volume of an ice block when heat is added from -200°c to 0°c... Will it volume increase or decrease?
adefenwa Reply
no
Emmanuel
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Kate
BPED..is my course.
Kate
No
Emmanuel
I think it is neither decreases nor increases ,it remains in the same volume because of its crystal structure
Sobirjon
100g of water is mixed with 60g of a liquid of relative density 1.2.assuming no changes in volume occurred,find the average relative density of the mixture. take density of water as 1g/cm3 and density of liquid as 1.2g/cm3
Lila
Sorry what does it means"no changes in volume occured"?
Sobirjon
volume can be the amount of space occupied by an object. But when an object does not change in shape it will still occupy the same space. Thats why the volume will still remain the same
Ben
Most soilds expand when heated but if it changes state at 0C it will have less volume. Ice floats because it is less dense ie a larger mass per unit volume.
Richard
how to calculate velocity
Okwethu Reply
v=d/t
Emeka
his about the speed?
Villaflor
how about speed
Villaflor
v=d/t
Nisha
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Jacob Reply
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Chase
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lawan Reply
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Okwethu
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akinmeji
Hello
Mishael
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Jerry
hi
Sakhi
hi
H.C
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akinmeji
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Ajayi
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ABDUL
If someone has not studied Mathematics enough yet, should theu study it first then study Phusics or Study Basics of Physics whilst srudying Math as well?
Riaz Reply
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Nuru
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Badmus
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Thomas
I guess that's it
Thomas
later people
Thomas
mathematics is everywhere
Anand
thanks but dat doesn't mean it is good without maths @Riaz....... Maths is essential in sciences particularly wen it comes to PHYSICS but PHYSICS must be started from the basic which may also help in ur mathematical ability
Nuru
A hydrometer of mass 0.15kg and uniform cross sectional area of 0.0025m2 displaced in water of density 1000kg/m3.what depth will the hydrometer sink
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16.66 meters?
Darshik
16.71m2
aways
,i have a question of let me give answer
aways
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aways
oscillation is a to and fro movement, it can also be referred to as vibration. e.g loaded string, loaded test tube or an hinged door
Olatunji Reply
what property makes the magnet to break?
Akshaya Reply
difference between charge and force
uwadiae Reply
charges: these are groups of ions which can be positive or negative. while force is said to be push or pull of an object. Generally, charge deals with ions while force deals with particles.
akinmeji
define oscillation
Munachukwu
what is up thrust
Lila Reply
upward force exerted by water on a body.
Matthew
A hydrometer of mass 0.15kg and of uniform cross sectional area 0.0025m2 displaced in water of density 1000kg/m3.what depth will the hydrometer sink
Lila
archimedes principle
Nicholas
mg=p(area x height)g
Nicholas
solve for height
Nicholas
what is matter
daniel Reply
Anything that has mass. And it occupies space. So it has volume.
wenhe
stages of matter.
daniel
States of matter are solid liquid gas and plasma
wenhe
what are Newtown's law
Lila
An object as rest will remain at rest or in motion will remain at motion. Force is mass times acceleration. And a force will have equal and opposite reaction.
wenhe
do we have any thing like plasma as a states of matter
daniel
no
Nicholas
Plasma is a state of matter. It's much later on. The sun for example is not a gas ball. Its plasma.
wenhe
why is plasma a state of matter
Lila
In lower grades you are taught that there are 3 states of matter. But later on, you'll learn that if you eat gas enough. It turns to plasma
wenhe
Heat*
wenhe
yes we have plasma as a state of matter
akinmeji
it is anything that occupie space and has mass
Anand
I never head of that before
daniel
Anything that occupies space and has mass is matter.
wenhe
what is the difference between vapor and gas
Lila
Plasma is best described as an ionized gas because it is made up of positively and negatively charged particles.
akinmeji
Vapour is gas.
wenhe
Aside plasma, we also have Bose- Einstein. this is also another state of matter
akinmeji
was bose Einstein discovered by Albert Einstein
Lila
We have like 6 states or more. But those are newly discovered.
wenhe
OK, thanks at list I have experience about that one now.
daniel
Einstein​ predicted it
wenhe
With another person
wenhe
what are the other states
Lila
Fermionic condensate
wenhe
OK what is Albert Einstein
daniel
And maybe more. I just rmb those. Tbh for school you only need to know 4.
wenhe
Albert Einstein​is the name of a scientist
wenhe
OK
daniel
what is the physical state of water and salt
Lila
can I ask off topic question?
Kristine
Both are matter. So can theoretically exist in all those states. But in room temperature, water is liquid and salt is solid.
wenhe
Yes
wenhe
Its about precalculus
Kristine
I'm not very good at maths. But if I can answer... Lol
wenhe
The reflector of a radiotelescope is in the shape of a parabola revolve its axis, if the diameter of the reflector is 400ft. above the vertex of the parabola, what should be the depth of reflector?
Kristine
Are there any other factors given?
wenhe
Or just that?
wenhe
matter is merely energy condensed to a slow vibration
Vashist
Cuz with those factors, I can't solve it.
wenhe
none
Kristine
Well. There must be another factor given or something. Like an equation of the parabola. Or else I can't solve it
wenhe
explain why water and salt are compounds
Lila
That's chem. But both water and salt are made of more than 1 element. So they are compounds.
wenhe
Compound meaning 2 or more things added together.
wenhe
salt chemically is combination of sodium n chloride whereas water is oxygen n hydrogen
Friday
deficiency of vitamin E cause ?
Friday
I don't know if this is the equation or formula your looking for; (x-h)²=4c(y-k) (x-h)²=-4c(y-k) (y-k)²=-4c(x-h) (y-k)²=4c(x-h)
Kristine
If the parabola opens downward, the equation will be (x-h)²=-4c(y-k). If the parabola opens upward, (x-h)²=4c(y-k). If the parabola opens to the left it will be -4c and if the parabola opens to the right it will be 4c
Kristine

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