<< Chapter < Page Chapter >> Page >

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

A car battery is rated to be able to move 600 ampere-hours (A·h) 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?


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 PE elec = qV size 12{"PE" rSub { size 8{"elec"} } = ital "qV"} {} , we have to calculate the charge q size 12{q} {} in 600 A·h , which is the product of the current I and the time t size 12{t} {} . We then multiply the result by 12.0 V. We can then calculate the battery’s increase in mass using Δ E = PE elec = ( Δ m ) c 2 size 12{ΔE="PE" rSub { size 8{"elec"} } = \( Δm \) c rSup { size 8{2} } } {} . Part (b) is a simple ratio converted to a percentage.

Solution for (a)

  1. Identify the knowns. I t = 600 A h ; V = 12 . 0 V size 12{V="12" "." 0`V} {} ; c = 3 . 00 × 10 8 m/s
  2. Identify the unknown. Δ m size 12{m} {}
  3. Choose the appropriate equation. PE elec = ( Δ m ) c 2
  4. Rearrange the equation to solve for the unknown. Δ m = PE elec c 2 size 12{m= { {"PE" rSub { size 8{"elec"} } } over {c rSup { size 8{2} } } } } {}
  5. Plug the knowns into the equation.
    Δ m = PE elec c 2 = qV c 2 = ( I t ) V c 2 = ( 600 A h ) ( 12.0 V ) ( 3.00 × 10 8 ) 2 .

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

    Δ m = ( 600 C/s h 3600 s 1 h ( 12.0 J/C ) ( 3.00 × 10 8 m/s ) 2 = ( 2.16 × 10 6 C ) ( 12.0 J/C ) ( 3.00 × 10 8 m/s ) 2

    Using the conversion 1 kg m 2 /s 2 = 1 J , we can write the mass as

    Δ m = 2.88 × 10 10 kg .

Solution for (b)

  1. Identify the knowns. Δ m = 2.88 × 10 10 kg ; m = 20.0 kg
  2. Identify the unknown. % change
  3. Choose the appropriate equation. % increase = Δ m m × 100% size 12{%" increase"= { {Δm} over {m} } times "100"%} {}
  4. Plug the knowns into the equation.
    % increase = Δ m m × 100% = 2.88 × 10 10 kg 20.0 kg × 100% = 1.44 × 10 9 % .


Both the actual increase in mass and the percent increase are very small, since energy is divided by c 2 size 12{c rSup { size 8{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 1 2 mv 2 size 12{ { {1} over {2} } ital "mv" rSup { size 8{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 net = KE . size 12{W rSub { size 8{"net"} } ="KE"} {}

Relativistically, at rest we have rest energy E 0 = mc 2 . The work increases this to the total energy E = γmc 2 . Thus,

Questions & Answers

Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Mykayuh Reply
Because I'm writing a report and I would like to be really precise for the references
Gre Reply
where did you find the research and the first image (ECG and Blood pressure synchronized)? Thank you!!
Gre Reply
Practice Key Terms 3

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Physics 101. OpenStax CNX. Jan 07, 2013 Download for free at http://legacy.cnx.org/content/col11479/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Physics 101' conversation and receive update notifications?