# 1.3 Gravitational potential energy  (Page 2/5)

 Page 2 / 5

## Converting between potential energy and kinetic energy

Gravitational potential energy may be converted to other forms of energy, such as kinetic energy. If we release the mass, gravitational force will do an amount of work equal to $\text{mgh}$ on it, thereby increasing its kinetic energy by that same amount (by the work-energy theorem). We will find it more useful to consider just the conversion of ${\text{PE}}_{\text{g}}$ to $\text{KE}$ without explicitly considering the intermediate step of work. (See [link] .) This shortcut makes it is easier to solve problems using energy (if possible) rather than explicitly using forces.

More precisely, we define the change in gravitational potential energy $\text{Δ}{\text{PE}}_{\text{g}}$ to be

$\text{Δ}{\text{PE}}_{\text{g}}=\text{mgh},$

where, for simplicity, we denote the change in height by $h$ rather than the usual $\text{Δ}h$ . Note that $h$ is positive when the final height is greater than the initial height, and vice versa. For example, if a 0.500-kg mass hung from a cuckoo clock is raised 1.00 m, then its change in gravitational potential energy is

$\begin{array}{lll}\text{mgh}& =& \left(\text{0.500 kg}\right)\left({\text{9.80}\phantom{\rule{0.25em}{0ex}}\text{m/s}}^{2}\right)\left(\text{1.00 m}\right)\\ & =& \text{4.90 kg}\cdot {m}^{2}{\text{/s}}^{2}\text{= 4.90 J.}\end{array}$

Note that the units of gravitational potential energy turn out to be joules, the same as for work and other forms of energy. As the clock runs, the mass is lowered. We can think of the mass as gradually giving up its 4.90 J of gravitational potential energy, without directly considering the force of gravity that does the work .

## Using potential energy to simplify calculations

The equation $\text{Δ}{\text{PE}}_{\text{g}}=\text{mgh}$ applies for any path that has a change in height of $h$ , not just when the mass is lifted straight up. (See [link] .) It is much easier to calculate $\text{mgh}$ (a simple multiplication) than it is to calculate the work done along a complicated path. The idea of gravitational potential energy has the double advantage that it is very broadly applicable and it makes calculations easier. From now on, we will consider that any change in vertical position $h$ of a mass $m$ is accompanied by a change in gravitational potential energy $\text{mgh}$ , and we will avoid the equivalent but more difficult task of calculating work done by or against the gravitational force.

## The force to stop falling

A 60.0-kg person jumps onto the floor from a height of 3.00 m. If he lands stiffly (with his knee joints compressing by 0.500 cm), calculate the force on the knee joints.

Strategy

This person’s energy is brought to zero in this situation by the work done on him by the floor as he stops. The initial ${\text{PE}}_{\text{g}}$ is transformed into $\text{KE}$ as he falls. The work done by the floor reduces this kinetic energy to zero.

Solution

The work done on the person by the floor as he stops is given by

$W=\text{Fd}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\theta =-\text{Fd},$

with a minus sign because the displacement while stopping and the force from floor are in opposite directions $\left(\text{cos}\phantom{\rule{0.25em}{0ex}}\theta =\text{cos}\phantom{\rule{0.25em}{0ex}}\text{180º}=-1\right)$ . The floor removes energy from the system, so it does negative work.

The kinetic energy the person has upon reaching the floor is the amount of potential energy lost by falling through height $h$ :

$\text{KE}=-\text{Δ}{\text{PE}}_{\text{g}}=-\text{mgh},$

The distance $d$ that the person’s knees bend is much smaller than the height $h$ of the fall, so the additional change in gravitational potential energy during the knee bend is ignored.

The work $W$ done by the floor on the person stops the person and brings the person’s kinetic energy to zero:

$W=-\text{KE}=\text{mgh}.$

Combining this equation with the expression for $W$ gives

$-\text{Fd}=\text{mgh}.$

Recalling that $h$ is negative because the person fell down , the force on the knee joints is given by

$F=-\frac{\text{mgh}}{d}=-\frac{\left(\text{60.0 kg}\right)\left(\text{9.80 m}{\text{/s}}^{2}\right)\left(-3\text{.}\text{00 m}\right)}{5\text{.}\text{00}×{\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}\text{m}}=3\text{.}\text{53}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}N.$

Discussion

Such a large force (500 times more than the person’s weight) over the short impact time is enough to break bones. A much better way to cushion the shock is by bending the legs or rolling on the ground, increasing the time over which the force acts. A bending motion of 0.5 m this way yields a force 100 times smaller than in the example. A kangaroo's hopping shows this method in action. The kangaroo is the only large animal to use hopping for locomotion, but the shock in hopping is cushioned by the bending of its hind legs in each jump.(See [link] .)

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
what is hormones?
Wellington
Got questions? Join the online conversation and get instant answers!