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Rotational inertia and moment of inertia

Before we can consider the rotation of anything other than a point mass like the one in [link] , we must extend the idea of rotational inertia to all types of objects. To expand our concept of rotational inertia, we define the moment of inertia     I size 12{I} {} of an object to be the sum of mr 2 size 12{ ital "mr" rSup { size 8{2} } } {} for all the point masses of which it is composed. That is, I = mr 2 size 12{I= Sum {} ital "mr" rSup { size 8{2} } } {} . Here I size 12{I} {} is analogous to m size 12{m} {} in translational motion. Because of the distance r size 12{r} {} , the moment of inertia for any object depends on the chosen axis. Actually, calculating I size 12{I} {} is beyond the scope of this text except for one simple case—that of a hoop, which has all its mass at the same distance from its axis. A hoop’s moment of inertia around its axis is therefore MR 2 size 12{ ital "MR" rSup { size 8{2} } } {} , where M size 12{M} {} is its total mass and R size 12{R} {} its radius. (We use M size 12{M} {} and R size 12{R} {} for an entire object to distinguish them from m size 12{m} {} and r size 12{r} {} for point masses.) In all other cases, we must consult [link] (note that the table is piece of artwork that has shapes as well as formulae) for formulas for I size 12{I} {} that have been derived from integration over the continuous body. Note that I size 12{I} {} has units of mass multiplied by distance squared ( kg m 2 size 12{"kg" cdot "m" rSup { size 8{2} } } {} ), as we might expect from its definition.

The general relationship among torque, moment of inertia, and angular acceleration is

net τ = size 12{τ=Iα} {}

or

α = net τ I , size 12{α= { { ital "net"τ} over {I} } ","} {}

where net τ size 12{τ} {} is the total torque from all forces relative to a chosen axis. For simplicity, we will only consider torques exerted by forces in the plane of the rotation. Such torques are either positive or negative and add like ordinary numbers. The relationship in τ = α = net τ I size 12{τ=Iα,`````α= { { ital "net"τ} over {I} } } {} is the rotational analog to Newton’s second law and is very generally applicable. This equation is actually valid for any torque, applied to any object, relative to any axis.

As we might expect, the larger the torque is, the larger the angular acceleration is. For example, the harder a child pushes on a merry-go-round, the faster it accelerates. Furthermore, the more massive a merry-go-round, the slower it accelerates for the same torque. The basic relationship between moment of inertia and angular acceleration is that the larger the moment of inertia, the smaller is the angular acceleration. But there is an additional twist. The moment of inertia depends not only on the mass of an object, but also on its distribution of mass relative to the axis around which it rotates. For example, it will be much easier to accelerate a merry-go-round full of children if they stand close to its axis than if they all stand at the outer edge. The mass is the same in both cases; but the moment of inertia is much larger when the children are at the edge.

Take-home experiment

Cut out a circle that has about a 10 cm radius from stiff cardboard. Near the edge of the circle, write numbers 1 to 12 like hours on a clock face. Position the circle so that it can rotate freely about a horizontal axis through its center, like a wheel. (You could loosely nail the circle to a wall.) Hold the circle stationary and with the number 12 positioned at the top, attach a lump of blue putty (sticky material used for fixing posters to walls) at the number 3. How large does the lump need to be to just rotate the circle? Describe how you can change the moment of inertia of the circle. How does this change affect the amount of blue putty needed at the number 3 to just rotate the circle? Change the circle’s moment of inertia and then try rotating the circle by using different amounts of blue putty. Repeat this process several times.

Questions & Answers

What is the difference between a principle and a law
the law is universally proved. The principal depends on certain conditions.
Dr
state Faraday first law
aliyu Reply
it states that mass of an element deposited during electrolysis is directly proportional to the quantity of electricity discharge
Olamide
what does the speedometer of a car measure ?
Jyoti Reply
Car speedometer measures the rate of change of distance per unit time.
Moses
describe how a Michelson interferometer can be used to measure the index of refraction of a gas (including air)
WILLIAM Reply
using the law of reflection explain how powder takes the shine off a person's nose. what is the name of the optical effect?
WILLIAM
is higher resolution of microscope using red or blue light?.explain
WILLIAM
what is dimensional consistent
Mohammed
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities and units of measure and tracking these dimensions as calculations or comparisons are performed
syed
can sound wave in air be polarized?
WILLIAM Reply
Unlike transverse waves such as electromagnetic waves, longitudinal waves such as sound waves cannot be polarized. ... Since sound waves vibrate along their direction of propagation, they cannot be polarized
Astronomy
A proton moves at 7.50×107m/s perpendicular to a magnetic field. The field causes the proton to travel in a circular path of radius 0.800 m. What is the field strength?
Celedonio Reply
derived dimenionsal formula
Ajak Reply
what is the difference between mass and weight
Isru Reply
assume that a boy was born when his father was eighteen years.if the boy is thirteen years old now, how is his father in
Isru
31yrs
Olamide
what is head-on collision
Javaid Reply
what is airflow
Godswill Reply
derivative of first differential equation
Haruna Reply
why static friction is greater than Kinetic friction
Ali Reply
draw magnetic field pattern for two wire carrying current in the same direction
Ven Reply
An American traveler in New Zealand carries a transformer to convert New Zealand’s standard 240 V to 120 V so that she can use some small appliances on her trip.
nkombo Reply
What is the ratio of turns in the primary and secondary coils of her transformer?
nkombo
what is energy
Yusuf
How electric lines and equipotential surface are mutually perpendicular?
Abid Reply
The potential difference between any two points on the surface is zero that implies È.Ŕ=0, Where R is the distance between two different points &E= Electric field intensity. From which we have cos þ =0, where þ is the angle between the directions of field and distance line, as E andR are zero. Thus
MAHADEV
sorry..E and R are non zero...
MAHADEV
Practice Key Terms 3

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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