# 0.3 Gravity and mechanical energy  (Page 8/9)

 Page 8 / 9

From this we see that when an object is lifted, like the suitcase in our example, it gains potential energy. As it falls back to the ground, it will lose this potential energy, but gain kinetic energy. We know that energy cannot be created or destroyed, but only changed from one form into another. In our example, the potential energy that the suitcase loses is changed to kinetic energy.

The suitcase will have maximum potential energy at the top of the cupboard and maximum kinetic energy at the bottom of the cupboard. Halfway down it will have half kinetic energy and half potential energy. As it moves down, the potential energy will be converted (changed) into kinetic energy until all the potential energy is gone and only kinetic energy is left. The $19,6\phantom{\rule{2pt}{0ex}}J$ of potential energy at the top will become $19,6\phantom{\rule{2pt}{0ex}}J$ of kinetic energy at the bottom.

During a flood a tree truck of mass $100\phantom{\rule{2pt}{0ex}}\mathrm{kg}$ falls down a waterfall. The waterfall is $5\phantom{\rule{2pt}{0ex}}m$ high. If air resistance is ignored, calculate

1. the potential energy of the tree trunk at the top of the waterfall.
2. the kinetic energy of the tree trunk at the bottom of the waterfall.
3. the magnitude of the velocity of the tree trunk at the bottom of the waterfall.

• The mass of the tree trunk $m=100\phantom{\rule{2pt}{0ex}}\mathrm{kg}$
• The height of the waterfall $h=5\phantom{\rule{2pt}{0ex}}m$ . These are all in SI units so we do not have to convert.
• Potential energy at the top
• Kinetic energy at the bottom
• Velocity at the bottom
1. $\begin{array}{ccc}\hfill PE& =& mgh\hfill \\ \hfill PE& =& \left(100\right)\left(9,8\right)\left(5\right)\hfill \\ \hfill PE& =& 4900\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}\hfill \end{array}$
2. The kinetic energy of the tree trunk at the bottom of the waterfall is equal to the potential energy it had at the top of the waterfall. Therefore $KE=4 900\phantom{\rule{2pt}{0ex}}J$ .

3. To calculate the velocity of the tree trunk we need to use the equation for kinetic energy.

$\begin{array}{ccc}\hfill KE& =& \frac{1}{2}m{v}^{2}\hfill \\ \hfill 4900& =& \frac{1}{2}\left(100\right)\left({v}^{2}\right)\hfill \\ \hfill 98& =& {v}^{2}\hfill \\ \hfill v& =& 9,899...\hfill \\ \hfill v& =& 9,90\phantom{\rule{0.166667em}{0ex}}\mathrm{m}·{\mathrm{s}}^{-1}\mathrm{downwards}\hfill \end{array}$

A $2\phantom{\rule{2pt}{0ex}}\mathrm{kg}$ metal ball is suspended from a rope. If it is released from point $A$ and swings down to the point $B$ (the bottom of its arc):

1. Show that the velocity of the ball is independent of it mass.
2. Calculate the velocity of the ball at point $B$ .

• The mass of the metal ball is $m=2\phantom{\rule{2pt}{0ex}}\mathrm{kg}$
• The change in height going from point $A$ to point $B$ is $h=0,5\phantom{\rule{2pt}{0ex}}m$
• The ball is released from point $A$ so the velocity at point, ${v}_{A}=0\phantom{\rule{2pt}{0ex}}m·s{}^{-1}$ .

All quantities are in SI units.

• Prove that the velocity is independent of mass.
• Find the velocity of the metal ball at point $B$ .
1. As there is no friction, mechanical energy is conserved. Therefore:

$\begin{array}{ccc}\hfill {U}_{A}& =& {U}_{B}\hfill \\ \hfill P{E}_{A}+K{E}_{A}& =& P{E}_{B}+K{E}_{B}\hfill \\ \hfill mg{h}_{A}+\frac{1}{2}m{\left({v}_{A}\right)}^{2}& =& mg{h}_{B}+\frac{1}{2}m{\left({v}_{B}\right)}^{2}\hfill \\ \hfill mg{h}_{A}+0& =& 0+\frac{1}{2}m{\left({v}_{B}\right)}^{2}\hfill \\ \hfill mg{h}_{A}& =& \frac{1}{2}m{\left({v}_{B}\right)}^{2}\hfill \end{array}$

As the mass of the ball $m$ appears on both sides of the equation, it can be eliminated so that the equation becomes:

$g{h}_{A}=\frac{1}{2}{\left({v}_{B}\right)}^{2}$
$2g{h}_{A}={\left({v}_{B}\right)}^{2}$

This proves that the velocity of the ball is independent of its mass. It does not matter what its mass is, it will always have the same velocity when it falls through this height.

2. We can use the equation above, or do the calculation from 'first principles':

$\begin{array}{ccc}\hfill {\left({v}_{B}\right)}^{2}& =& 2g{h}_{A}\hfill \\ \hfill {\left({v}_{B}\right)}^{2}& =& \left(2\right)\left(9.8\right)\left(0,5\right)\hfill \\ \hfill {\left({v}_{B}\right)}^{2}& =& 9,8\hfill \\ \hfill {v}_{B}& =& \sqrt{9,8}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}·{\mathrm{s}}^{-1}\hfill \end{array}$

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
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Source:  OpenStax, Maths test. OpenStax CNX. Feb 09, 2011 Download for free at http://cnx.org/content/col11236/1.2
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