# 8.5 Dynamics of rotational motion: rotational inertia  (Page 4/8)

 Page 4 / 8

Torque is the analog of force and moment of inertia is the analog of mass. Force and mass are physical quantities that depend on only one factor. For example, mass is related solely to the numbers of atoms of various types in an object. Are torque and moment of inertia similarly simple?

No. Torque depends on three factors: force magnitude, force direction, and point of application. Moment of inertia depends on both mass and its distribution relative to the axis of rotation. So, while the analogies are precise, these rotational quantities depend on more factors.

## Section summary

• The farther the force is applied from the pivot, the greater is the angular acceleration; angular acceleration is inversely proportional to mass.
• If we exert a force $F$ on a point mass $m$ that is at a distance $r$ from a pivot point and because the force is perpendicular to $r$ , an acceleration $\text{a = F/m}$ is obtained in the direction of $F$ . We can rearrange this equation such that
$\mathrm{F = ma}\text{,}$

and then look for ways to relate this expression to expressions for rotational quantities. We note that $\mathrm{a = r\alpha }$ , and we substitute this expression into $\mathrm{F=ma}$ , yielding

$\mathrm{F=mr\alpha }$
• Torque is the turning effectiveness of a force. In this case, because $F$ is perpendicular to $r$ , torque is simply $\tau =\mathit{rF}$ . If we multiply both sides of the equation above by $r$ , we get torque on the left-hand side. That is,
$\text{rF}={\text{mr}}^{2}\alpha$

or

$\tau ={\text{mr}}^{2}\alpha \text{.}$
• The moment of inertia $I$ of an object is the sum of ${\text{MR}}^{2}$ for all the point masses of which it is composed. That is,
$I=\sum {\text{mr}}^{2}\text{.}$
• The general relationship among torque, moment of inertia, and angular acceleration is
$\tau =\mathrm{I\alpha }$

or

$\begin{array}{}\alpha =\frac{\text{net τ}}{I}\cdot \\ \end{array}$

## Conceptual questions

The moment of inertia of a long rod spun around an axis through one end perpendicular to its length is ${\mathit{ML}}^{2}\text{/3}$ . Why is this moment of inertia greater than it would be if you spun a point mass $M$ at the location of the center of mass of the rod (at $L/2$ )? (That would be ${\mathit{ML}}^{2}\text{/4}$ .)

Why is the moment of inertia of a hoop that has a mass $M$ and a radius $R$ greater than the moment of inertia of a disk that has the same mass and radius? Why is the moment of inertia of a spherical shell that has a mass $M$ and a radius $R$ greater than that of a solid sphere that has the same mass and radius?

Give an example in which a small force exerts a large torque. Give another example in which a large force exerts a small torque.

While reducing the mass of a racing bike, the greatest benefit is realized from reducing the mass of the tires and wheel rims. Why does this allow a racer to achieve greater accelerations than would an identical reduction in the mass of the bicycle’s frame?

A ball slides up a frictionless ramp. It is then rolled without slipping and with the same initial velocity up another frictionless ramp (with the same slope angle). In which case does it reach a greater height, and why?

## Problems&Exercises

This problem considers additional aspects of example Calculating the Effect of Mass Distribution on a Merry-Go-Round . (a) How long does it take the father to give the merry-go-round and child an angular velocity of 1.50 rad/s? (b) How many revolutions must he go through to generate this velocity? (c) If he exerts a slowing force of 300 N at a radius of 1.35 m, how long would it take him to stop them?

(a) 0.338 s

(b) 0.0403 rev

(c) 0.313 s

what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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.
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
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
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
Do you know which machine is used to that process?
s.
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
how to synthesize TiO2 nanoparticles by chemical methods
Zubear
what's the program
Jordan
?
Jordan
what chemical
Jordan
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
Got questions? Join the online conversation and get instant answers!