# 8.7 Angular momentum and its conservation  (Page 2/7)

 Page 2 / 7

## Calculating the torque in a kick

The person whose leg is shown in [link] kicks his leg by exerting a 2000-N force with his upper leg muscle. The effective perpendicular lever arm is 2.20 cm. Given the moment of inertia of the lower leg is $1.25 kg\cdot {\text{m}}^{2}$ , (a) find the angular acceleration of the leg. (b) Neglecting the gravitational force, what is the rotational kinetic energy of the leg after it has rotated through $\text{57}\text{.}3º$ (1.00 rad)?

Strategy

The angular acceleration can be found using the rotational analog to Newton’s second law, or $\alpha =\text{net}\phantom{\rule{0.25em}{0ex}}\tau /I$ . The moment of inertia $I$ is given and the torque can be found easily from the given force and perpendicular lever arm. Once the angular acceleration $\alpha$ is known, the final angular velocity and rotational kinetic energy can be calculated.

Solution to (a)

From the rotational analog to Newton’s second law, the angular acceleration $\alpha$ is

$\alpha =\frac{\text{net}\phantom{\rule{0.25em}{0ex}}\tau }{I}.$

Because the force and the perpendicular lever arm are given and the leg is vertical so that its weight does not create a torque, the net torque is thus

$\begin{array}{lll}\text{net}\phantom{\rule{0.25em}{0ex}}\tau & =& {r}_{\perp }F\\ & =& \left(0\text{.}\text{0220 m}\right)\left(\text{2000}\phantom{\rule{0.25em}{0ex}}\text{N}\right)\\ & =& \text{44}\text{.}\text{0 N}\cdot \text{m.}\end{array}$

Substituting this value for the torque and the given value for the moment of inertia into the expression for $\alpha$ gives

$\alpha =\frac{\text{44}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{N}\cdot \text{m}}{1\text{.}\text{25}\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot {\text{m}}^{2}}=\text{35}\text{.}2\phantom{\rule{0.25em}{0ex}}{\text{rad/s}}^{2}.$

Solution to (b)

The final angular velocity can be calculated from the kinematic expression

${\omega }^{2}={{\omega }_{0}}^{2}+2\text{αθ}$

or

${\omega }^{2}=2\text{αθ}$

because the initial angular velocity is zero. The kinetic energy of rotation is

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

so it is most convenient to use the value of ${\omega }^{2}$ just found and the given value for the moment of inertia. The kinetic energy is then

$\begin{array}{lll}{\text{KE}}_{\text{rot}}& =& 0.5\left(1\text{.25}\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot {\text{m}}^{2}\right)\left(\text{70.}4\phantom{\rule{0.25em}{0ex}}{\text{rad}}^{2}/{\text{s}}^{2}\right)\\ & =& \text{44}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{J}\end{array}.$

Discussion

These values are reasonable for a person kicking his leg starting from the position shown. The weight of the leg can be neglected in part (a) because it exerts no torque when the center of gravity of the lower leg is directly beneath the pivot in the knee. In part (b), the force exerted by the upper leg is so large that its torque is much greater than that created by the weight of the lower leg as it rotates. The rotational kinetic energy given to the lower leg is enough that it could give a ball a significant velocity by transferring some of this energy in a kick.

## Making connections: conservation laws

Angular momentum, like energy and linear momentum, is conserved. This universally applicable law is another sign of underlying unity in physical laws. Angular momentum is conserved when net external torque is zero, just as linear momentum is conserved when the net external force is zero.

## Conservation of angular momentum

We can now understand why Earth keeps on spinning. As we saw in the previous example, $\text{Δ}L=\left(\text{net}\phantom{\rule{0.25em}{0ex}}\tau \right)\text{Δ}t$ . This equation means that, to change angular momentum, a torque must act over some period of time. Because Earth has a large angular momentum, a large torque acting over a long time is needed to change its rate of spin. So what external torques are there? Tidal friction exerts torque that is slowing Earth’s rotation, but tens of millions of years must pass before the change is very significant. Recent research indicates the length of the day was 18 h some 900 million years ago. Only the tides exert significant retarding torques on Earth, and so it will continue to spin, although ever more slowly, for many billions of years.

Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
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
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
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
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
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.
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!