# 9.4 Rotational kinetic energy: work and energy revisited  (Page 4/9)

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Solution for (a)

The rotational kinetic energy is

${\text{KE}}_{\text{rot}}=\frac{1}{2}{\mathrm{I\omega }}^{2}.$

We must convert the angular velocity to radians per second and calculate the moment of inertia before we can find ${\text{KE}}_{\text{rot}}$ . The angular velocity $\omega$ is

$\omega =\frac{\text{300 rev}}{\text{1.00 min}}\cdot \frac{\text{2π rad}}{\text{1 rev}}\cdot \frac{\text{1.00 min}}{\text{60.0 s}}=\text{31.4}\frac{\text{rad}}{\text{s}}.$

The moment of inertia of one blade will be that of a thin rod rotated about its end, found in [link] . The total $I$ is four times this moment of inertia, because there are four blades. Thus,

$I=4\frac{{\mathrm{M\ell }}^{2}}{3}=4×\frac{\left(\text{50.0 kg}\right){\left(\text{4.00 m}\right)}^{2}}{3}=\text{1067 kg}\cdot {\text{m}}^{2}.$

Entering $\omega$ and $I$ into the expression for rotational kinetic energy gives

$\begin{array}{lll}{\text{KE}}_{\text{rot}}& =& 0.5\left(\text{1067 kg}\cdot {\text{m}}^{2}\right){\left(\text{31.4 rad/s}\right)}^{2}\\ & =& 5.26×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{J}\end{array}$

Solution for (b)

Translational kinetic energy was defined in Uniform Circular Motion and Gravitation . Entering the given values of mass and velocity, we obtain

${\text{KE}}_{\text{trans}}=\frac{1}{2}{\mathit{mv}}^{2}=\left(0.5\right)\left(\text{1000 kg}\right){\left(\text{20.0 m/s}\right)}^{2}=2\text{.}\text{00}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{J}.$

To compare kinetic energies, we take the ratio of translational kinetic energy to rotational kinetic energy. This ratio is

$\frac{2\text{.}\text{00}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{J}}{5\text{.}\text{26}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{J}}=0.380.$

Solution for (c)

At the maximum height, all rotational kinetic energy will have been converted to gravitational energy. To find this height, we equate those two energies:

${\text{KE}}_{\text{rot}}={\text{PE}}_{\text{grav}}$

or

$\frac{1}{2}{\mathrm{I\omega }}^{2}=\text{mgh}.$

We now solve for $h$ and substitute known values into the resulting equation

$h=\frac{{\frac{1}{2}\mathrm{I\omega }}^{2}}{\text{mg}}=\frac{5.26×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{J}}{\left(\text{1000 kg}\right)\left(9.80\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}\right)}=\text{53.7 m}.$

Discussion

The ratio of translational energy to rotational kinetic energy is only 0.380. This ratio tells us that most of the kinetic energy of the helicopter is in its spinning blades—something you probably would not suspect. The 53.7 m height to which the helicopter could be raised with the rotational kinetic energy is also impressive, again emphasizing the amount of rotational kinetic energy in the blades. The first image shows how helicopters store large amounts of rotational kinetic energy in their blades. This energy must be put into the blades before takeoff and maintained until the end of the flight. The engines do not have enough power to simultaneously provide lift and put significant rotational energy into the blades. The second image shows a helicopter from the Auckland Westpac Rescue Helicopter Service. Over 50,000 lives have been saved since its operations beginning in 1973. Here, a water rescue operation is shown. (credit: 111 Emergency, Flickr)

## Making connections

Conservation of energy includes rotational motion, because rotational kinetic energy is another form of $\text{KE}$ . Uniform Circular Motion and Gravitation has a detailed treatment of conservation of energy.

## How thick is the soup? or why don’t all objects roll downhill at the same rate?

One of the quality controls in a tomato soup factory consists of rolling filled cans down a ramp. If they roll too fast, the soup is too thin. Why should cans of identical size and mass roll down an incline at different rates? And why should the thickest soup roll the slowest?

The easiest way to answer these questions is to consider energy. Suppose each can starts down the ramp from rest. Each can starting from rest means each starts with the same gravitational potential energy ${\text{PE}}_{\text{grav}}$ , which is converted entirely to $\text{KE}$ , provided each rolls without slipping. $\text{KE}$ , however, can take the form of ${\text{KE}}_{\text{trans}}$ or ${\text{KE}}_{\text{rot}}$ , and total $\text{KE}$ is the sum of the two. If a can rolls down a ramp, it puts part of its energy into rotation, leaving less for translation. Thus, the can goes slower than it would if it slid down. Furthermore, the thin soup does not rotate, whereas the thick soup does, because it sticks to the can. The thick soup thus puts more of the can’s original gravitational potential energy into rotation than the thin soup, and the can rolls more slowly, as seen in [link] .

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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
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da
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
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Application of nanotechnology in medicine
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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
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Rafiq
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Damian
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LITNING
scanning tunneling microscope
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Rafiq
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Mahi
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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
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write examples of Nano molecule?
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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
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what king of growth are you checking .?
Renato
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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